Houston, who will be out of contract in the summer, met with the club's board on Tuesday after being granted permission to speak to Championship side Blackpool over replacing Michael Appleton.
Dundee United chairman Stephen Thompson said on Monday that Houston could stay at the club, although he added: "But only if he accepts my terms."
A new statement on the club website read: "Directors of Dundee United met with manager Peter Houston and his representative to discuss his contract situation.
"These discussions were amicable and will continue later this week."
Houston had earlier admitted he was surprised to be given the all-clear to speak to Blackpool as he had yet to conclude contract talks with United.
He said: "It's unexpected. I haven't applied for any jobs. We were still in negotiations over a new deal for me and my staff.
"Mr Thompson and myself's working relationship recently has been fine.
"What he has stated is he has to make cutbacks in staff costs. Before the contact from Blackpool he did say he didn't want to hold me back from bettering myself.
"If he thinks that then I'm grateful to him for allowing me to speak to them. Nothing has been decided because Blackpool still have to get in touch with me."