With Steve Agnew still in caretaker control for this afternoon's Championship game at Norwich City, Gibson will finalise a four-man shortlist over the weekend.
The name of bookmakers' favourite Tony Mowbray will be included on the shortlist, with two other former Middlesbrough captains Paul Ince and Nigel Pearson also understood to be featuring prominently in Gibson's thoughts.
The final candidate is likely to be either Gary Megson, Steve Round or Phil Brown, although a number of highprofile figures have applied for the job and their credentials will be considered in the next 48 hours.
Former Boro midfielder Graeme Souness is not a serious contender, having privately ruled himself out of the running this week.
Gibson conceded that the club got it wrong by appointing Strachan when he appeared on BBC Tees on Monday.
The Boro chairman is determined to leave no stone unturned in an attempt to ensure that the Scotsman's successor is the right man for the job.
A small section of Middlesbrough supporters have questioned his continued commitment in the wake of a poor start to the season, a surprisingly short-sighted viewpoint given a 25-year involvement that has seen Gibson steer Boro to their first major trophy and oversee the development of both the Riverside Stadium and Rockliffe Park training complex.
The Boro chief remains as driven and ambitious as ever, and McMahon is confident he will play a pivotal role in turning the club's fortunes around.
I've heard some people questioning him, which is quite bizarre because for as long as I can remember he's done everything right here and I think he'll continue to do that for a long time, said the defender.
McMahon is one of the club's longest-serving players, having skippered the Academy side to FA Youth Cup success in 2004.
You can't avoid all the speculation as to who might come in.
There are names flying around but it's not our job to get involved with that.
The chairman's got to do his job now and appoint who he thinks is the best man.
Tony Mowbray It's the name that's going around.
But the chairman will do what's best for the club, as he always does.
I'm sure the chairman will do his best for Middlesbrough in making his appointment.
In the meantime, Agnew will attempt to prevent Boro slipping into the bottom three when they travel to Carrow Road this afternoon The former first-team coach was immediately installed after Strachan tendered his resignation on Monday afternoon, a development that caught the club's players by surprise.
Strachan had overseen training on Monday, and had shown no outward indication of what was to come.
Boro's players left the training ground as usual after lunch, and spent the afternoon exchanging text messages and watching the breaking news channels as details of Strachan's departure began to emerge.
The last thing we saw of him was running up the hill after training, said McMahon.
There was no sign at all that anything was wrong.
The last contact I had with him was a bit of a laugh and a joke. I took a bit of a dive in training, and he told me I'd done the right thing by diving because he didn't want me getting injured.
That's what he's like. He's a bright, bubbly character and very funny.
He was good with all the players. He's a little bit sarcastic, but that's not having a go at anyone, it's just the way he is. You just have to know how to take him.
We've tried to go about things like we do every day, but as players we feel partly responsible for him going.
He wasn't a bad manager, it was just one of those things that didn't work out.
Strachan has left Boro much closer to the bottom three than the top six, but despite the club having failed to win any of their last five matches, McMahon insists there are reasons to remain optimistic with 34 games left.
No-one wants to think about the worst that can happen, we've got to be positive and start looking up, he said. At the minute, we're at the wrong end, looking down.
Results haven't been good, but confidence isn't low.
The league table doesn't look good at the minute, but a win will get us looking up instead of looking down. If we keep creating chances, I'm sure we'll start getting goals.
But we have to get a win desperately.
That's why this is such a big game.