Swansea have rejected £40m offers from Everton and Leicester for the Iceland international, whom they value at £50m.
"You do live in hope that he will stay," Britton told BBC Radio 5 live.
"I don't know if it's going to get done. If he does go, it's vitally important to replace him with the same kind of quality."
Sigurdsson scored nine goals last season as the Welsh club managed to avoid relegation from the Premier League.
The 27-year-old did not travel with Swansea for their tour of the United States and was also left out of Saturday's 2-0 pre-season win at Birmingham City because of the transfer speculation.
The Swans are being linked with Las Palmas midfielder Jonathan Viera, who would be a potential replacement should Sigurdsson depart.
"It's a difficult one, it has maybe dragged on longer than a lot of people have expected," Britton, 34, said.
"Gylfi wasn't in America, he didn't travel with the squad for Birmingham away. When things like that happen you do think that 'yes, the transfer will happen'.
"Things do change overnight, maybe Everton move on to another target they think is better value for money, maybe a new target becomes available.
"But it would be a massive blow, you can't deny the influence he had on the team.
"When you see Gylfi's stats last year you think he would be playing for one of the top-half clubs, he is a big influence at the top end of the pitch.
"Unfortunately at the kind of club we're at, when the biggest teams come calling it is hard to keep him."
Leon Britton says Swansea City must invest in their playing squad if his fellow midfielder Gylfi Sigurdsson leaves the club.