Tottenham were thought to be favourites ahead of Newcastle to sign the 28-year-old but Boro chief executive Keith Lamb has revealed they have also accepted an offer from another, as yet unnamed side.
"We've agreed a fee with Tottenham and as far as I'm aware he went down at the weekend to have a medical and we're just waiting to see," Lamb said.
"We agreed a fee with Tottenham, Newcastle and another club but we left it up to Jonathan, he could stay or go."
Woodgate is one of Middlesbrough's prize assets but the player has been troubled by injuries in the last few years and Lamb said manager Gareth Southgate was happy to let the player go providing his valuation - believed to be around £8million - was met.
Arsenal have been linked this morning with a rival offer for Woody.