West Ham had appeared in pole position, but reports suggest Spurs' late move may pay off.
Speaking after Tuesday's 2-0 win over Fulham, Spurs boss Harry Redknapp said he expected Gudjohnsen to join West Ham but confirmed they had made an enquiry about taking him on loan for the rest of the season, with his parent club Monaco paying the bulk of the wages.
"I spoke to his agent, he said he was going to West Ham, so I left it with the agent," Redknapp said.
"I would take him on a loan. He would be good loan signing playing in different systems.
"He's a good footballer and an interesting player on a loan but I think he's gone to West Ham."