Woosnam captained Europe to victory at the 2006 Ryder Cup in Ireland
Europe will name a Ryder Cup captain for 2010 on 29 January, after failing to come to a decision on Tuesday.
The Ryder Cup committee met for almost four hours in Abu Dhabi, but will reconvene on 28 January and plan to name their choice the following day.
The favourites to lead Europe when Wales hosts the tournament at Celtic Manor are Jose Maria Olazabal, Ian Woosnam and Sandy Lyle.
The United States have named former US Open champion Corey Pavin as captain.
Under Paul Azinger, the US regained the Ryder Cup last year after three successive victories for Europe.
Ireland's three-time major champion Padraig Harrington told BBC Radio 5 Live he sees the stand-out candidates as Lyle and Olazabal but admitted it would be a tough decision.
"I think Jose is going to do a great job when it comes around to him being captain and there's no doubt everybody is looking forward to him being captain," said Harrington.
"But the fact that he's thrown in there is confusing the issue for Sandy, maybe.
"Sandy has been a stalwart in Europe for a long number of years and in many ways deserves the Ryder Cup captaincy.
"Maybe because he's a reasonably quiet individual and hasn't lobbied so much in the past and maybe because he's drifted away, purely due to age, from the younger players on tour.
"But certainly if it was a few years ago I don't think we'd be having this discussion, Sandy Lyle would probably be the Ryder Cup captain."
Former Ryder Cup-winning captain Sam Torrance has backed Woosnam, who skippered the side to victory in Ireland in 2006, to succeed Nick Faldo as Europe's captain.
Torrance told BBC Radio 5 Live: "We just lost so it's imperative we put the right man in there.
"I like Ian Woosnam. Woosie did it in Ireland, but this is Wales.
"I don't think anybody can bring the Welsh support like Ian could.
Olazabal, captain of the European team beaten by Asia in the Royal Trophy in Bangkok at the weekend, gave the strongest hint yet that he would be willing to take on the captaincy.
"I would love to be considered, I'm not going to deny that," he said. "I would love to be playing and I think maybe it is too soon, but time will tell."
One man who has backed the 42-year-old is another former captain, Bernard Gallacher, but he believes Olazabal's desire to continue playing could hamper his chances.
"It's not straightforward because the committee said it's a non-playing appointment," Gallacher told 5 Live.
"That makes it difficult as far as Olazabal is concerned. He seems to feel he can make the team, he's still got ambitions, so we'll have to wait and see.
"I feel Olazabal should say to himself, 'I am not going to make this team, I'm going to accept this appointment', and that would be the easy thing to do."
Sergio Garcia also suggested Olazabal would prefer to play in Europe's 2010 rather than take on the captain's role.
"There's no doubt he would make a great captain, but we have been talking about it and he has told me he wants to make that team as a player," said Garcia.
"If he stays healthy he still has a chance of getting on the team and I would love to have him there as a team-mate."
Tony Jacklin, who captained Europe to Ryder Cup victory in 1985 and the first ever European victory in the US in 1987, has also backed Olazabal for the role, but has not ruled out Lyle leading Europe.
"I certainly think he (Olazabal) would be a deserved captain and we're about due for one of the Europeans to come on board as captain," he said.
"He's made great contributions, not only to the game of golf, but to the Ryder Cup itself.
"Sandy's also got good credentials, he's done a lot of for European golf and we tend to forget how brilliant his golf was in the 1980s.
"He's obviously in there as a candidate, but ultimately the decision rests with the committee and I'm sure they'll put their heads together and make the correct decision."