Swansea are reportedly holding out for £50 million for the playmaker, who scored nine goals in the Premier League last season to help Swansea to survive in the top-flight.
But Clement indicated there has been progress in the negotiations with Everton and a deal could now be completed before the weekend.
"Talks are still going on and hopefully we are going to get some news in the next couple of days," Clement said.
"The deal needs to be the best for all parties involved, we have got a very, very important season coming up. We know what happened last season if you don't make a good start and we don't want that to happen again.
"We want to make a good start and everyone wants to know what our squad is going to be going into the first game of the season."
Ashley Williams swapped Swansea for Everton in the week before the start of the last Premier League season, with the club subsequently winning only one of their first 12 games of the campaign, a 1-0 success at Burnley on the opening day.
With Everton seemingly unwilling to meet Swansea's £50m valuation for Sigurdsson, a player-plus-cash deal has been mooted and Clement is open to such a deal proving the transfer is completed quickly.
"Assuming it is the right player coming our way and it is someone we are interested in and fills a need in our squad, those deals will always have potential," Clement said.
"I hope it is done today, either way, whether that is selling or not, but we need a quick resolution to the situation."