As a franchised service though? Manchester have contracted out everything run from a GM depot, but all of their cross boundary routes do have a significant part within Greater Manchester. There isn't much reason for Merseytravel to care about the 375/385, and if they choose not to franchise them there won't be any commercial operators nearby to take them on.
I would guess they will be franchised, but Merseytravel becoming the dominant operator in West Lancashire creates obvious potential for arguments in future about whether West Lancashire residents should have a say in how the buses are run, or whether Merseyside taxpayers money should be used to run buses in Lancashire.
There's also the question of the 319 between Skelmersdale and Kirkby. Run for the benefit of Lancashire residents, subsidised by Lancashire County Council, currently run from Gillmoss. Will that arrangement be allowed to continue? I think it was said that operators of large franchise depots in Manchester aren't allowed to use those depots to bid for other work. (Edited)