Gudjohnsen arrives on loan from Monaco, Wilson comes in on a permanent basis from Portsmouth and Pennant will stay on loan from Real Zaragoza until January, the club announced on their website.
The club are awaiting confirmation from the Football Association and Premier League that the transfers were completed in time.
The arrival of former Chelsea and Barcelona forward Gudjohnsen, 31, for the season would be a particular coup for Potters boss Tony Pulis.
Ex-Liverpool winger Pennant, 27, agreed a loan deal to stay at the Britannia Stadium until January after a troubled year in Spain.
Portsmouth's Republic of Ireland Under-21 international Wilson, 23, joined in a player-plus-cash move that saw midfielder Liam Lawrence, 28, and striker Dave Kitson, 30, move in the opposite direction.
Wilson, who can play in either midfield or defence, signed a four-year contract.
Wilson said on his new club's official website: "This is a great club which is on the up and so this is a positive move for me.
"I had a taste of playing in the Premier League last season; it's the best league in the world and I wanted to get back there as soon as possible.
"I spoke to (Potters coach) Mark O'Connor when I knew Stoke were interested in me because he was responsible for bringing me through as a youngster at Portsmouth. He's a great guy and he has certainly been a big influence upon me choosing to come here."
Pulis said: "I wanted to freshen the squad up with some young players, and Marc certainly fits that criteria.
"He showed great maturity last season in what was a very difficult time for Portsmouth, so he will give us more competition for midfield places."
He added: "The deal has enabled Lawrence and Kitson to move on, giving them a fresh start in their careers and the chance of regular first-team football.
"We would like to thank them for their contribution and wish them all the best. Liam, in particular, has been a great servant of this club and he has played a major part in the success that we have achieved over the past three seasons."
Source: Team Talk
Source: Team Talk