The news is sure to spark renewed interest from a host of Premiership clubs for the pounds 5m-rated former Blackpool winger. Leeds, Sheffield Wednesday, Blackburn and Tottenham have been linked with Sinclair.
Houston said that the Loftus Road board would consider Sinclair's request at their next meeting, probably later this week.
"I don't want him to go and I have had a long conversation with him, but the player has obviously thought long and hard about it and has put his request in writing," he said.
Rangers gained their second successive win under Houston, but were made to hang on as Tranmere rallied well in the closing stages.The game was riddled with encounters in the heavy conditions, but Rangers got the best possible start with Steve Slade heading home from a Trevor Sinclair cross in the ninth minute.
Tranmere drew level after 41 minutes thanks to a penalty by John Aldridge, his 10th goal of the season. He fooled goalkeeper Danny Coyne from the spot with his characteristic shimmy after he had been brought down by Alan McDonald who was fortunate to escape with only a booking. The defender stayed on the field and made his presence felt after 66 minutes with a glancing header from a Sinclair corner to regain the lead.
Rangers continued to have the better of things at this stage and looked to have sewn the game up with five minutes remaining when substitute Lee Charles scored after a good run by Sinclair. However, Ged Brannan gave the Merseysiders hope with a well-struck right-foot shot from a corner just minutes later.
"We have had a hard demanding week and done very well with seven points out of nine,'' Houston said.
Aldridge, the Tranmere player-manager, said: "We were punished for three defensive mistakes. "It is a worrying situation for us and I will have to sit down and look at it in the cold light of day. We tried hard at times but we just weren't good enough."
Goals: Slade (9) 0-1; Aldridge (pen 41) 1-1; McDonald (67) 1-2; Charles (85) 1-3; Brannan (87) 2-3.
Tranmere Rovers: Coyne; Stevens, Brannan, Higgins, Teale, O'Brien, Morrissey (Cook, 78), Aldridge, Irons (Mahon, 68), Branch, Bonetti (Nevin, 54).
Queen's Park Rangers: Sommer; Graham, Brevett, Barker, McDonald, Ready, Brazier, Murray, Dichio (Charles, 58), Slade (Quashie, 77), Sinclair. Substitutes not used: Plummer.
Bookings: QPR: McDonald, Ready.
Referee: K Lynch (Knaresborough).
Man of the match: Sinclair.
Attendance: 7,025.Reuse content