Gündogan, 32, is out of contract with City at the end of June, but the Champions League winners have offered him the chance to extend his seven-year stay at the club.
Barca are also keen to sign the Germany international, and Guardiola said he knows the Catalan club's coach, Xavi Hernández, has been in touch with the player.
"We are interested in renewing his contract and [Barcelona] are also interested," Guardiola told reporters at the Puma Legends golf tournament on Monday.
"I hope he stays with us. We are still fighting for him to stay with us. He is a player we want to stay -- and we will do all we can -- but we know Barca want him.
"If Barca sign him, they will have a spectacular footballer on their hands. I am aware Xavi has called him a lot. If, in the end, he picks Barca, I will tell him he will enjoy himself."
Xavi has made it clear he wants to add more goals and creativity to midfield this summer, saying a "Santi Cazorla-type player" would be the ideal signing to help build on a first LaLiga title since 2019.
Xavi has said Barca were also keen on signing Cancelo in January, but that City would not sanction the Portugal international's move to Spotify Camp Nou.
"I read that, but Xavi has been misinformed," Guardiola said in response to those comments. "The reality is that João had to leave and we didn't mind where he went -- no offer arrived from Barca."
Mbappé said he still wants to remain at PSG next season, but sources have told ESPN that to avoid losing him for free, the French club will look to move him on this summer.
"We won't make a move for Mbappé," the City coach confirmed. "You already know where he wants to go."