Stevenson’s form at the Tynecastle club earned him a call up to the Scotland squad for a game against Cyprus before he left Edinburgh for Partick Thistle in 2014 where played 47 times.
He returned to Ayr last December on loan, with the move made permanent earlier this year when he was released by the Firhill side. And Stevenson helped Ayr to a second promotion via the play-offs last month.
Ayr manager Ian McCall was keen to keep him on board for life in the Championship but Dumbarton manager Stephen Aitken is confident that he will get his man.
Aitken said: “Ryan is a very experienced player and one that I would be delighted to bring to Dumbarton. We have had an encouraging chat and I reckon that we could get him on board.
“I will know for definite in the next 24 hours.”
Dumbarton are also considering a move for former Hearts winger David Smith who was released last week after two years at Falkirk.
Smith, 23, played more than 50 games for the Bairns but dropped out of manager Peter Houston’s plans in the second half of last season.