Several national media reports have claimed that the deal was worth £45 million.
Swansea have claimed for the past few weeks that they will only consider bids of £50 million for the attacking midfielder.
Sigurdsson has been part of transfer speculation for a long time, and this summer he has been linked with Leicester City, and more heavily with Everton.
Local media reported that Swansea turned down a bid of £40 million from Leicester for Sigurdsson, and a bid of similar territory from Everton.
Now the Merseyside club have come back with a second offer worth £45 million, which appears to have been swiftly rejected by the Welsh side.
Several media outlets claimed that the offer was of £40 million plus an additional £5 million in add-ons.
Sigurdsson did not travel with the rest of the squad on their United States tour, claiming that he was not in “the right frame of mind to travel.”
The squad has now returned from their tour, and it will certainly be interesting to see whether Sigurdsson is involved in their next pre-season game against Birmingham City on Saturday July 29th.
What is going to happen next?
Swansea manager Paul Clement hasn’t ruled out his playmaker staying at the club. He said: “I don’t think it’s impossible (Sigurdsson will stay),”
"And I think from all sides we want to see a resolution to this situation."
He added: "I spoke to Gylfi two days ago.
"I have a very good relationship with Gylfi and the chairman and the owners know very much my feeling on the situation and we're talking daily about how things are moving.”
Swansea chairman Huw Jenkins also stated that the club would do everything in their power to keep hold of their star players, meaning Sigurdsson, Fernando Llorente and Alfie Mawson at the start of the window.
It was thought that Swansea and Everton would meet in the middle following the latter’s previous £40 million bid for Sigurdsson, but Swansea appear to be holding firm on their £50 million valuation of the Icelandic international.