The 'Gravity' singer - who dumped the Hollywood actress in March this year - reportedly telephoned her in tears and begged to be given another chance.
A source said: "John asked Jen if they could go back to seeing each other exclusively. The call came as a massive surprise and of course Jen is delighted with John's announcement - who wouldn't be?"
The 31-year-old star has promised to give the relationship his "best shot" if she agrees to take him back.
John even hinted he wanted them to "build a life together" and possibly get married.
The source explained to Britain's Look magazine: "John suggested that if all is going well with them in six months, then they can begin discussing marriage and building a life together. Jen was floored! John always seemed so anti-commitment before. But now he's begging Jen for another chance and he's promising to give the relationship his best shot.
"Despite his big plans, John suggested they don't rush into moving in together. She'll keep her house in Beverly Hills, while John will continue to live in Pacific Palisades. I think it's just because he wants to move slowly and avoid mistakes they made previously."
Since their split, Jennifer has been linked to a string of other stars.
She reportedly dated Gerard Butler - who she starred alongside in the upcoming comedy 'The Bounty' and was spotted enjoying an intimate dinner date with her 'He's Just Not That Into You' co-star Bradley Cooper.
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