Coach Dominic Kinnear on Wednesday said Akinbiyi, 34, is set to undergo a physical this weekend in Houston.
The much-travelled forward, 34, is out of contract at Turf Moor in the summer and has made just one league start this season.
Clarets boss Owen Coyle said: "Ade is in negotiations with the MLS over a move to Houston and he would go there with our blessing.
"He has been a magnificent professional and a great servant to Burnley Football Club. But he wants a change of lifestyle and I would never stand in his way."
Agent Gary Mellor added: "We're in negotiations with the MLS over a move and we are hopeful of reaching an agreement.
"Ade will meet the Houston manager this weekend and we will take things from there."