diff --git a/src/__mocks__/tsconfig-src.json b/src/__mocks__/tsconfig-mocks.json
similarity index 100%
rename from src/__mocks__/tsconfig-src.json
rename to src/__mocks__/tsconfig-mocks.json
diff --git a/src/config/config-set.spec.ts b/src/config/config-set.spec.ts
index 821ee61c96..dc72b77b64 100644
--- a/src/config/config-set.spec.ts
+++ b/src/config/config-set.spec.ts
@@ -51,7 +51,7 @@ describe('parsedTsConfig', () => {
   })
 
   it('should fallback to ES2015 as default target and CommonJS as default module when no target or module defined in tsconfig', () => {
-    const compilerOptions = get({ tsconfig: 'tsconfig.spec.json' }).options
+    const compilerOptions = get({ tsconfig: 'src/__mocks__/tsconfig-mocks.json' }).options
 
     expect(compilerOptions.target).toBe(ts.ScriptTarget.ES2015)
     expect(compilerOptions.module).toBe(ts.ModuleKind.CommonJS)
diff --git a/src/utils/__snapshots__/backports.spec.ts.snap b/src/utils/__snapshots__/backports.spec.ts.snap
index 46276af0ff..b681d85848 100644
--- a/src/utils/__snapshots__/backports.spec.ts.snap
+++ b/src/utils/__snapshots__/backports.spec.ts.snap
@@ -233,6 +233,35 @@ Array [
 ]
 `;
 
+exports[`backportJestConfig with "globals.ts-jest.tsConfig" set to 'tsconfig.build.json' should have changed the config correctly: before 1`] = `
+Object {
+  "globals": Object {
+    "ts-jest": Object {
+      "tsConfig": "tsconfig.build.json",
+    },
+  },
+}
+`;
+
+exports[`backportJestConfig with "globals.ts-jest.tsConfig" set to 'tsconfig.build.json' should have changed the config correctly: migrated 1`] = `
+Object {
+  "globals": Object {
+    "ts-jest": Object {
+      "tsconfig": "tsconfig.build.json",
+    },
+  },
+}
+`;
+
+exports[`backportJestConfig with "globals.ts-jest.tsConfig" set to 'tsconfig.build.json' should warn the user 1`] = `
+Array [
+  "[level:40] \\"[jest-config].globals.ts-jest.tsConfig\\" is deprecated, use \\"[jest-config].globals.ts-jest.tsconfig\\" instead.
+",
+  "[level:40] Your Jest configuration is outdated. Use the CLI to help migrating it: ts-jest config:migrate <config-file>.
+",
+]
+`;
+
 exports[`backportJestConfig with "globals.ts-jest.tsConfigFile" set to 'tsconfig.build.json' should have changed the config correctly: before 1`] = `
 Object {
   "globals": Object {
diff --git a/src/utils/backports.spec.ts b/src/utils/backports.spec.ts
index 675ba482e0..1dfe486116 100644
--- a/src/utils/backports.spec.ts
+++ b/src/utils/backports.spec.ts
@@ -1,7 +1,7 @@
 import { inspect } from 'util'
 
 import { testing } from 'bs-logger'
-import set = require('lodash/set')
+import set from 'lodash/set'
 
 import { backportJestConfig } from './backports'
 
@@ -13,20 +13,23 @@ beforeEach(() => {
 })
 
 describe('backportJestConfig', () => {
-  // eslint-disable-next-line @typescript-eslint/no-explicit-any
-  const makeTestsFor = (oldPath: string, _: string, values: any[]) => {
+  const makeTestsFor = (oldPath: string, values: unknown[]) => {
     values.forEach((val) => {
       // eslint-disable-next-line @typescript-eslint/no-explicit-any
       let original: any
+
       beforeEach(() => {
         original = {}
         set(original, oldPath, val)
       })
+
       describe(`with "${oldPath}" set to ${inspect(val)}`, () => {
         it('should warn the user', () => {
           backportJestConfig(logger, original)
+
           expect(logTarget.lines.warn).toMatchSnapshot()
         }) // should warn the user
+
         it('should have changed the config correctly', () => {
           expect(original).toMatchSnapshot('before')
           expect(backportJestConfig(logger, original)).toMatchSnapshot('migrated')
@@ -35,17 +38,19 @@ describe('backportJestConfig', () => {
     }) // for
   } // makeTestsFor
 
-  makeTestsFor('globals.__TS_CONFIG__', 'globals.ts-jest.tsConfig', [{ foo: 'bar' }])
+  makeTestsFor('globals.__TS_CONFIG__', [{ foo: 'bar' }])
+
+  makeTestsFor('globals.__TRANSFORM_HTML__', [true, false])
 
-  makeTestsFor('globals.__TRANSFORM_HTML__', 'globals.ts-jest.stringifyContentPathRegex', [true, false])
+  makeTestsFor('globals.ts-jest.tsConfigFile', ['tsconfig.build.json'])
 
-  makeTestsFor('globals.ts-jest.tsConfigFile', 'globals.ts-jest.tsConfig', ['tsconfig.build.json'])
+  makeTestsFor('globals.ts-jest.tsConfig', ['tsconfig.build.json'])
 
-  makeTestsFor('globals.ts-jest.enableTsDiagnostics', 'globals.ts-jest.diagnostics', [true, false, '\\.spec\\.ts$'])
+  makeTestsFor('globals.ts-jest.enableTsDiagnostics', [true, false, '\\.spec\\.ts$'])
 
-  makeTestsFor('globals.ts-jest.useBabelrc', 'globals.ts-jest.babelConfig', [true, false])
+  makeTestsFor('globals.ts-jest.useBabelrc', [true, false])
 
-  makeTestsFor('globals.ts-jest.typeCheck', 'globals.ts-jest.isolatedModules', [true, false])
+  makeTestsFor('globals.ts-jest.typeCheck', [true, false])
 
-  makeTestsFor('globals.ts-jest.skipBabel', 'globals.ts-jest.babelConfig', [true, false])
+  makeTestsFor('globals.ts-jest.skipBabel', [true, false])
 })
diff --git a/src/utils/backports.ts b/src/utils/backports.ts
index 2b0085fa40..3887c42616 100644
--- a/src/utils/backports.ts
+++ b/src/utils/backports.ts
@@ -62,6 +62,14 @@ export const backportJestConfig = <T extends Config.InitialOptions | Config.Proj
     delete tsJest.tsConfigFile
   }
 
+  if ('tsConfig' in tsJest) {
+    warnConfig('globals.ts-jest.tsConfig', 'globals.ts-jest.tsconfig')
+    if (tsJest.tsConfig) {
+      mergeTsJest.tsconfig = tsJest.tsConfig
+    }
+    delete tsJest.tsConfig
+  }
+
   if ('enableTsDiagnostics' in tsJest) {
     warnConfig('globals.ts-jest.enableTsDiagnostics', 'globals.ts-jest.diagnostics')
     if (tsJest.enableTsDiagnostics) {
diff --git a/tsconfig.build.json b/tsconfig.build.json
index 79255ad49b..77df3cf0cb 100644
--- a/tsconfig.build.json
+++ b/tsconfig.build.json
@@ -1,7 +1,6 @@
 {
   "extends": "./tsconfig.json",
   "compilerOptions": {
-    "target": "ES5",
     "sourceMap": false,
     "inlineSources": false,
     "inlineSourceMap": false,
diff --git a/tsconfig.json b/tsconfig.json
index 4f7f4b7a1c..5e2e081db3 100644
--- a/tsconfig.json
+++ b/tsconfig.json
@@ -1,6 +1,7 @@
 {
   "compilerOptions": {
     "module": "CommonJS",
+    "target": "ES5",
     "declaration": false,
     "noEmit": true,
     "downlevelIteration": true,