Council leader Sir Richard Leese has written to the Prime Minister to say they would only consider a joint bid.
Durban was due to be the first African city to host the Games, but was stripped of the status in March.
Liverpool, London and Birmingham have expressed an interest in hosting.
There have also been bids from Australia, Canada and Malaysia.
In the letter to Theresa May, the council leader said that Manchester is "prepared to do all we reasonably can to make our contribution and to ensure our technical capability is available to ensure that England are placed in the best possible position to host the 2022 Games".
In February, South Africa's sports minister Fikile Mbalula first indicated that Durban - awarded hosting rights in December 2015 - might not be able to arrange the 2022 event.
And in March, the Commonwealth Games Federation (CGF) decided the city did not meet the necessary criteria and said it would be looking for a new host city.
CGF chief David Grevemberg has said officials are looking to make a decision quickly, adding that a joint bid would be considered.
The Commonwealth Games are held every four years and feature athletes from more than 50 countries, mostly former British colonies.
Britain last hosted the Games in Glasgow in 2014, while the 2002 event was staged in Manchester. The next edition takes place on Australia's Gold Coast in 2018.
Manchester will not bid for the 2022 Commonwealth Games, with the city council deeming it "inappropriate" for the 2002 hosts to compete against other English cities for the event.