Johnson quit his post four days ago following a dismal World Cup campaign on and off the pitch.
Several people have already been linked with the job, including New Zealand's World Cup-winning supremo Graham Henry, former South Africa coach Nick Mallett and Northampton chief Jim Mallinder.
But Bath boss and former British and Irish Lions coach McGeechan feels time is on Twickenham's side, despite the RBS 6 Nations campaign starting in little more than two months' time.
"I will be very interested to see which direction they go," he said, following Bath's 16-13 Heineken Cup victory over Montpellier at the Recreation Ground.
"They should ideally look for an English coach. And I think it is key they probably take a bit of time to look at the best options."
Asked if a caretaker coach should initially be appointed, McGeechan added: "I think the only way it would work is if they keep the same coaches for six months and then have a six-month process for finding the right man.
"That could be better than jumping to conclusions in three or four weeks."
McGeechan looked on while World Cup winner Stephen Donald took centre stage for Bath as they put their European campaign back on track with a tense victory over dangerous French opposition.
The New Zealander, whose penalty secured World Cup final glory for the All Blacks against France in Auckland last month, marked his home debut in style.
Donald created Bath's opening try and then almost scored the second following a lengthy interception sprint, while he also slotted two penalties to get his team up and running in Heineken Cup Pool Three.
After losing away to Glasgow last weekend, Bath knew they could not afford a second successive defeat and effectively risk pool stage elimination, a fate seemingly destined for their fellow English representatives Northampton, Gloucester and London Irish.
Donald, who kicked two penalties, said: "We knew if we didn't get the result today, we would be making up the numbers.
"We played poorly in the second half, but they are a very good French team and we can look at it positively in that we hung on.
"Learning how to win these kind of games is half the thing of it. It was a tough loss against Glasgow last weekend - we stuffed it up.
"I love the concept of the Heineken Cup. We play Leinster next, and what is not to get excited about playing against the defending champions and half the Irish team? I can't wait for it."
Although Bath took charge early on, with wing Olly Woodburn and prop David Flatman scoring tries, Montpellier hit back through the efforts of their Argentinian wing Martin Bustos Moyano.
He scored a try and added eight points from the boot, hauling Montpellier back into contention after they trailed by 13 points early in the second half.
McGeechan added: "We showed in the first half the rugby we can play. Some poor execution let them in a bit, but the character the players showed in the second half pleased me.
"We didn't panic as a team and we worked well under pressure. We looked two yards quicker as a team than we were last week, which got us the advantage we needed at key times in the first half.
"Leinster will be tough. They are used to winning in Europe now, they won pretty comfortably (against Glasgow) today, and we know it will be a really good challenge for us."
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