But McGeechan can still send out a unit comprising England's Joe Worsley, Ireland openside Jonny O'Connor and highly-rated prospect James Haskell, with Dallaglio on the bench, where Rees could join him.
"The back-row is such a competitive area at the moment, and we are faced with some difficult choices each week," McGeechan said.
"Since coming back from injury, Lawrence has played more full games than perhaps we anticipated, and the plan was always to review his progress after three or four weeks.
"Consequently, this week, Lawrence will take his turn on the bench.
"We really are spoiled for choice in the back-row, with the likes of Dallaglio, O'Connor, Worsley, Rees and Haskell all fit and available."
Wasps, still in the Premiership's top four despite a narrow defeat against champions Sale Sharks last time out, should be too strong for an erratic Bath side who include ex-Leeds Rhinos rugby league star Chev Walker among their replacements.
Leicester, meanwhile, must overcome the loss of 10 players to international duty as they face Saracens at Vicarage Road on Sunday.
Martin Corry, Lewis Moody, Julian White, George Chuter, Ben Kay, Marcos Ayerza, Martin Castrogiovanni, Jim Hamilton, Geordan Murphy and Seru Rabeni are all absent, but Tigers' strength in depth means they can still travel south with
England international half-backs Andy Goode and Harry Ellis among their replacements.
"The physio and medical teams have done a great job," said Leicester head coach Pat Howard.
"The only two players who are not available because of injury are Henry Tuilagi and Ollie Smith, and it is a pretty similar line-up to last week."
Saracens have put England World Cup winner Richard Hill on stand-by for the game, although he is not currently in their match-day 22.
If Hill receives a late call-up, it would be his first senior appearance since suffering a serious knee injury during last year's Lions tour of New Zealand.
"We go into the Leicester game expecting a very hard match," said Saracens rugby director Alan Gaffney, who hands former Great Britain rugby league captain Andy Farrell a start in midfield alongside Kevin Sorrell.
"Like us, they are missing players on international duty, but their result against Worcester last weekend showed they have enormous strength in depth, and that is something we are aiming for as a club."