The 40-year-old Coldplay frontman is believed to be romancing the 'Fifty Shades of Grey' star, and it has now been claimed that their relationship is set to progress to the next level, as he wants to introduce her to his former parter, and the mother of his two children, Apple, 13, and Moses, 11.
A source said: ''It was only a matter of time until they'd meet and Chris has already started suggesting an introduction.''
But it's believed that Dakota, 28, is ''intimidated'' by the prospect of meeting Gwyneth - who ended her 13-year marriage to Chris in 2016 - because the 'Fix You' hitmaker ''still puts her on a pedestal''.
The insider added to Grazia magazine: ''It's a slightly intimidating situation for Dakota - she's over a decade younger than Chris and Gwyneth - and her friends have joked that's it's like preparing to meet your mother-in-law for the first time.
''Chris still puts Gwyneth on a pedestal, and her opinion means a lot to him. It might be awkward, but Dakota may well find that she has to get used to it.''
It comes after 45-year-old Gwyneth recently revealed she sees her former spouse as a brother-like figure, because they've remained close since their split in 2014.
She said: ''He's really like my brother, we're very familial.
''It's nice, it's great ... Divorce is terrible, it was very painful, it was really hard. We really genuinely wanted our kids to be as unscathed as possible.
''We thought if we could really maintain the family even though we weren't a couple that was kind of the goal so that's what we've tried to do.''
Since splitting from Chris, the 'Iron Man' actress has struck up a romance with 46-year-old writer and producer Brad Falchuk, whom she is now engaged to.
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