Durban was stripped of hosting the Games in March after failing to meet the criteria set by the Commonwealth Games Federation (CGF).
Liverpool has declared its interest in staging the Games, as part of a bid to host the 2026 event.
Birmingham, London and Manchester have also expressed an interest in hosting.
"Hosting the Commonwealth Games in 2022 could give us an opportunity to step up and show the very best of global Britain as an outward-looking nation that is confident on the world stage," said Culture Secretary Karen Bradley.
"It's only right that we look into the potential for hosting one of the world's biggest sporting events on home soil."
The government will consider multi-city proposals and decide whether to submit a formal bid to the CGF later this year.
CGF chief David Grevemberg said officials were looking to make a decision quickly and would consider a joint bid.
"The London 2012 Olympic and Paralympic Games and Glasgow 2014 Commonwealth Games showed the appetite and enthusiasm that the British people have for supporting major sporting events," said Sports Minister Tracey Crouch.
"We will weigh up whether the UK should step in to host, looking at the opportunities to promote global Britain and deliver a lasting legacy at good value for taxpayers."
Britain hosted the 2014 Commonwealth Games in Glasgow, while the 2002 event was staged in Manchester.
British cities interested in hosting the 2022 Commonwealth Games have been asked by the UK government to submit proposals for staging the event.