The 37-year-old was on Thursday adamant the Teessiders have received no bids for any member of their current squad amid suggestions that Tottenham and Newcastle are ready to step up their respective interest in Downing and Woodgate.
The Boro chief, who is currently trying to make his own transfer breakthrough with Brazilian strikers Afonso Alves and Fred among his targets, has spent much of the last few weeks dealing with rumours that Downing in particular could leave the club.
He said: "It suggests to me we have got some good players. It doesn't annoy me because there is nothing you can do about it.
"Teams will speculate, teams might try to bid for your players - as yet, they haven't - but you get enquiries about your players all the time.
"In the end, we have the decisions and we have the say as to whether we decide to keep or sell players.
"It certainly doesn't annoy me. You look at your squad of players and if quite a few of them are attracting interest, that's a good sign, I would certainly say that.
"You look at certain other teams and say who would you take, and maybe there aren't too many.
"It certainly doesn't concern me at all."