Wiktionary
vb. 1 To snuggle. 2 To come physically close to, using body language in an attempt to persuade another (often hesitant) party to snuggle or embrace. 3 (by extension) To form a relationship with some one or some thing for the purpose of obtaining some benefit.
Usage examples of "cosy up".
He would be a Moczar man but that didn't tell me anything because the Minister of the Interior was the head of the police power and he could be in any one of its divisions, but he worried me because they're always on the lookout for exchange material and this was a suitable time: an international conference tends to cosy up the atmosphere and that's when people like fringe syndicate journalists step out of line and get hooked, in addition to which we had Blok and Shelepov in the Scrubs and they were both valuable enough for Moscow to try setting up arrangements.
But they probably figure that the way to counter a crazy man like George is to cosy up to him.
It was as if Burns knew the child before him was destined to outsell him, knew the runt would get a knighthood, build Abbotsford and cosy up to the King.