The estimated cost for a new 60,000 seat Stamford Bridge has increased to £1bn after delays, which included a dispute with a local family.
Abramovich is unwilling to invest in a major project in a country where he is not allowed work.
The 51-year-old's UK investor visa expired some weeks ago.
But it is understood that Abramovich's decision will have no impact on the running of the football team.
The delay in issuing him a new visa comes amid increased diplomatic tensions between London and Moscow after the poisoning of former Russian spy Sergei Skripal in Salisbury.
The British government has refused to comment on his individual case.
Chelsea issued a statement saying the club had delayed work on Stamford Bridge because of "the current unfavourable investment climate".
It added: "No further pre-construction design and planning work will occur. The club does not have a time frame set for reconsideration of its decision."
Plans were under consideration to move Chelsea matches to Wembley stadium for the four years it would take to redevelop Stamford Bridge.
Although the future of Wembley is in doubt after Fulham owner Shahid Khan offered £600m to buy the stadium from the Football Association, it is not thought to be a factor in Chelsea's decision.