According to ESPN, citing club sources - also reported by the BBC - Chelsea are considering a groundshare with West Ham at their new location at the Olympic Stadium during the upcoming three-year construction of Stamford Bridge. This would be an alternative option to the previously rumored Wembley and Twickenham temporary moves.
Wembley is still believed to be Chelsea's favoured option, even though rivals Tottenham Hotspur are reportedly closing in on a deal to use the stadium as their temporary home ground for the 2017-18 season. Chelsea vehemently opposed any groundshare with Spurs previously, though perhaps because of potential delays, this may not be an issue after all.
Twickenham would be a closer option, and without any groundshare concerns, but it's also the least likely thanks to heavy opposition from the locals and their council over congestion and noise.
West Ham's sweetheart deal with the London Legacy Development Corporation (which is not without its fair share of controversy of course) could prove financially beneficial to Chelsea as well, should we choose to go that route. The FA will be charging £15-20m per year for Wembley, but West Ham are only paying £2.5m for the use of the Olympic Stadium. Chelsea would presumably cover part of that rent under such an agreement.
The planning application for the new Stamford Bridge is still making the bureaucratic rounds of the Hammersmith & Fulham Council, and this process is expected to last beyond the end of the current season. Latest reports also warn that the project might experience delays due to issues with the railways and other concerned parties.
If the project does get delayed, Chelsea could have exclusive use of Wembley starting in 2018-19. But if the project stays on schedule, Chelsea sharing space with another football club seems inevitable. So the question is, more expensive Wembley (and Spurs for one of the three years) or less expensive Olympic Stadium (but three years of West Ham)?