Adam Reach scored against his old club in the 20th minute to lift the Owls nine points clear of the relegation zone after making it two wins in a row since Jos Luhukay’s exit last week.
Caretaker Bullen has overseen both those victories but it is understood former Sheffield United, Huddersfield Town and Hull City manager Steve Bruce is the favourite to take over.
“I would think there will be an announcement in the next couple of days. It is not the sort of thing you want dragging on and dragging on,” said Bullen.
“I think it needs to be done sooner rather than later so the players know exactly where they stand as well because you don’t want people in limbo.
“The sooner it is done the better, in my opinion, especially with a transfer window coming up. I don’t know where we are at financially. That’s not my remit but there is the potential to freshen things up.”
Wednesday looked comfortable for long periods as Middlesbrough struggled again at home.
And Bullen hailed the performance of his players after watching them beat promotion-chasing Middlesbrough 1-0 at the Riverside.
Bullen said: “Boro put us under a lot of pressure in the second half. We had to stand up well to stay disciplined and communicate well to withstand the onslaught.
“We struggled first five minutes but went into a diamond two and got a foothold in the game. After the first 10 minutes I felt we dominated the first half.
“In the second half, credit to Boro, they adjusted, came out with high energy. But for all the ball I can’t remember our keeper having a big save to make. The boy Lewis Wing skipped one past the post late on but as a coach you have to be delighted.”
It was the seventh time this season Boro have not won on home soil, leaving them nine points behind the top two a year on from Pulis taking over on Teesside.
The home fans booed the final whistle and at during the game sections chanted “attack, attack, attack” sarcastically after Wednesday had gone ahead.
Pulis said: “It’s frustrating. We had a good result at Reading, deserved the win there then come back here and ...
“In the first 10 minutes, quarter of an hour we started really bright, moved the ball around well, got round the back of them.
“Then they get a goal from their first effort, the disappointing thing was one or two dropped their heads a bit, which went through to half-time. I had a good chat with them.
“Whether you like it or not, you’ve got home support who are going to get behind you if you do things well, if you don’t they’re going to let you know, you have to accept that. The supporters of this club - it’s a big club - expect and demand a certain level.”
Apart from Reach’s goal, Wednesday did not have another effort on target.
Pulis said: “The problem is the final third. We’ve played a game of football again, we’ve got 30 crosses into the box, more possession, more set plays, but it falls down in final third. I think we need a couple of wins to dust everything down.
“The away form has been excellent but it’s been here that’s cost us.
“We need to keep the lads going as much as they can. They’re frustrated as much as anything else.”
Middlesbrough: Randolph, Ayala, Flint, Fry (Hugill 46), McNair, Howson (Wing 78), Clayton (Tavernier 68), Besic, Friend, Downing, Assombalonga. Unused substitutes: Konstantopoulos, Braithwaite, Saville, Walker.
Sheffield Wednesday: Westwood, Palmer, Lees, Hector, Fox, Hutchinson (Pelupessy 73), Matias, Reach, Bannan, Lucas Joao (Onomah 14, Boyd 66), Fletcher. Unused substitutes: Thorniley, Nuhiu, Baker, Dawson.
Referee: P Bankes (Merseyside).