Blues boss Ronald Koeman is keen to bolster his midfield before the window closes and has already been heavily linked with Manchester United's Morgan Schneiderlin.
United are already thought to have rejected one bid in the region of £19million and it remains to be seen if Koeman will be willing to match their £25million valuation.
The Dutchman is looking at alternatives and Swans star Siigurdsson has now emerged as a possible target as he continues to shine in a struggling side at the Liberty Stadium.
Leicester and Southampton are also keen with some reports in Iceland claiming the latter could offer as much as £34million to take him to St Mary's.
However Sigurdsson only recently penned a new deal to keep him at the club until 2020 and Swans officials have issued a hands-off warning, insisting he is not for sale.
bet365 World Cup 2018 Penalty Payback: Back a team to win the tournament and get a refund if your selection is eliminated on penalties. Applies to losing pre-match Win and Each Way bets on the To Win Outright market for World Cup 2018. Bet restrictions and T&Cs apply. New and eligible customers only.