The club announced a deal to acquire land at Bramley Moore Dock on which to build the new Â£300m ground on Thursday.
Mayor Joe Anderson said talks about using the stadium, which will seat 50,000, will happen "in due course".
Liverpool has expressed an interest in hosting the event in 2022 after the original hosts Durban pulled out.
It follows the city's announcement in 2016 that it would bid to host the 2026 Commonwealth Games.
Source: Museum of Liverpool
Mr Anderson said in a report to the city council's cabinet, who are due to discuss a deal which would see the authority acting as guarantor for the football club on Friday, that the new offer would be an "accelerated" version of that plan.
He added that the club could submit a planning application by the end of 2017.
A council spokesman said the new ground would be the city's Commonwealth Games stadium, but further details were not available as "at the moment, we are just discussing the deal".
Bramley Moore Dock was announced as the preferred site for the club's proposed new stadium in January.
Two other Premier League sides have taken up residence at stadiums built for international multi-sport events in recent years - Manchester City, whose Etihad Stadium was part of the 2002 Commonwealth Games, and West Ham United, who moved to the stadium used in the 2012 Olympics.
Red or Blue, most Liverpool residents will be behind the idea of regenerating North Liverpool's docklands.
If a new stadium can do for Kirkdale Ward what Manchester's Commonwealth Games stadium did for East Manchester, far more people will benefit from the plan than football fans.
Taken alongside the council's ambitious plans for what it calls the "Ten Streets" - essentially the roads shooting off the main thoroughfare to Bootle - this could be transformative for the area.
It is already coming alive with art. The influential gang behind The Kazimier recently opened their new venue nearby, and there are tech start-ups and social enterprises springing up amongst the pubs and garages.
But, Liverpool has seen a few false dawns when it comes to massive regeneration projects - and stadiums - and there is the small issue of decent transport links to sort out before hoards of sports fans can be welcomed.
Everton's new stadium will be a "key part" of Liverpool's bid to host the 2022 Commonwealth Games, the city's mayor has said.