“It’s fair to say we have tried to get this deal sorted in a short space of time, since we have come back to negotiating,” said Williams.
“I think that is evident and I think everyone is happy with that, because it is so important to have Liam’s contract situation sorted out.
“He is a player who gives everything for the club every single day, and you can see that on the pitch.”
In all, Pembrokeshire-born Cullen has scored 23 goals in 130 senior appearances since coming through the youth ranks at Swansea.
Cullen is one of a number of Swansea players whose contracts had been due to expire at the end of the season, with centre-backs Ben Cabango and Harry Darling among the others.
Cullen is the leading scorer in a Swansea side who have scored the fewest goals in the Championship this season.
“We have not been as potent as we would like to be, but we know having the likes of Liam Cullen will help us do that in the future,” Williams added.
“He is tireless, he gets through so much work for the team, but we know he has quality as well.”
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