The couple, who got engaged in June last year, could relocate to the Californian city if Lampard moves to LA Galaxy, David Beckham's football club.
Dancing on Ice host Bleakley, meanwhile, is said to be confident of finding presenting work in the US and "would not think twice" about moving there.
"They love LA and the trip in May is about testing the water to see if they could make it their home. They are very open to the idea," a friend told The Mail On Sunday.
"Frank and Christine have secretly been harbouring this fantasy of moving to America for a while, but nothing concrete was agreed until now. They are really excited at the possibility.
"They are flying to LA after Christine finishes the current series of Dancing on Ice and Frank has some time free between matches so they can check out possible job offers."
Lampard's agent denied claims he is planning a move to LA Galaxy after his contract with Chelsea ends in 2013, although the footballer is thought to have spoken with Beckham to get advice.
"Christine has no current plans to move to LA and is thoroughly enjoying Dancing on Ice as well as looking forward to upcoming new projects with ITV," Bleakley's spokesperson added.