The Evening Post reports that Swansea City are keen on the former Bolton, Chelsea, Stoke City, Tottenham and Fulham striker as one of their 'big name' signings of the summer.
Gudjohnsen is a free agent this summer and wanted by Fulham and other Premier League clubs whilst he has worked with Brendan when he was at Chelsea.
The Post reports It is understood Rodgers has spoken to Gudjohnsen's camp and that Swansea are now awaiting a response from the Icelandic international.