The 25-year old winger has been in talks with the Dynamo regarding a winter shift and according to Hearts, he has left Edinburgh to link up with an American team.
"Hearts can reveal that Andrew Driver has left the club to pursue his career in the United States," read a club statement.
"Andrew has made 28 appearances in a Hearts shirt this season, scoring twice. He leaves Gorgie with a Scottish Cup winner's medal, following his impressive display in the 2012 final, when Hearts brushed aside city rivals Hibs 5-1.
"The club wishes Andrew all the best as he continues his career with this fantastic opportunity to play in America."
Dynamo president Chris Canetti indicated that negotiations were well underway but stopped short of saying a deal had been done.
"I can confirm the reports out of the UK that we are in talks with Hearts and Andrew Driver on a possible move to the Dynamo," Canetti told MLSsoccer.com. "The process is still evolving, so that's all I can say at this time."