"Plymouth have made an inquiry but it has been turned down at this moment in time," said City boss Peter Reid.
However with the Icelander stuck in the reserves, his Highfield Road future looks bleak.
Gudjonsson impressed during a trial last season and was signed on a free from German side Bochum, but has had to contend with accusations from City supporters that he had played for a contract as he failed to show the same sort of form in pre and early season matches this term.
He was slated for poor performances during the first three league games of the season and was subsequently dropped by Peter Reid to the bench. His last first team outing was the League Cup tie at Middlesbrough.