Green’s Dictionary of Slang

to flatter, to curry favour, to talk nonsense in a soothing, plausible manner, esp. for the purposes of charming a woman one wishes to seduce.

1941		New Statesman 30 Aug. 218/3: [list of war slang] Flannel – To flatter.
1957		J. Braine Room at Top 46: I managed to flannel him into the belief that I approved of his particular brand of efficiency.
1962		R. Cook Crust on its Uppers 121: Just flannelled the old ice-cream a bit.
1970		Sun. Times 15 May 53: Some of the cast I suspect were flannelling, but there are two spellbinding performances by Ian Hogg and Pat Hartley.
1999		K. Sampson Powder 48: It was good to hear Guy simplify it like this, and good to know that he wasn’t flannelling them.