The Tannadice club are expected to unveil the former Celtic and Aberdeen player as their new manager on Wednesday.
McNamara, 39, has steered the Jags to second place in the Scottish First Division in his second season at Firhill.
The Thistle post is his first management position after a lengthy playing career with Dunfermline, Celtic, Wolves, Aberdeen, Falkirk, and Partick.
Houston announced two weeks ago that he would be leaving United at the end of the season, but brought forward his departure by leaving the club on Monday.
Falkirk boss Steven Pressley had been widely tipped to replace him.
Thistle confirmed they had granted their boss permission to speak to the Tayside club. A statement from Firhill read: "Following an approach from Dundee United, the board of directors of Partick Thistle Football Club have reluctantly granted permission for Jackie McNamara and Simon Donnelly to speak to them about their vacant managerial position."