Rangers have also made a fresh offer for the 22-year-old after having earlier bids rejected, but they have not matched the package on the table from Rovers. Goodwillie was left out the United squad to face Hearts in the Clydesdale Bank Premier League on Sunday.
"It's no secret Blackburn are interested in him. The clubs have agreed a fee," Houston told BBC Radio Scotland's Sportsound programme.
"David Goodwillie is likely to go if he can agree terms."
The fee is reported to be £2million up front and another £800,000 of potential add-ons, with a sell-on clause included.
Houston added: "The chairman is happy with the offer Blackburn have made. I think Rangers have made an improved offer but he hasn't said to me whether he's happy with that or not.
"It's between the two clubs just now but certainly the only club that has spoken to David is Blackburn Rovers, and David's advisers haven't agreed a fee yet. I don't know what the Rangers offer is. I have heard it's very similar but only the add-ons from Blackburn are better, that's all I know."
Goodwillie scored 19 goals for United last season and made his Scotland debut in a friendly win over the Faroe Islands in November.
The striker's Scotland career was put on hold after he was charged with a serious sexual assault in January, but the charge was dropped last week because of insufficient evidence.
That development sparked increased activity on the transfer front for Goodwillie, who has two years left on his Tannadice contract. He now looks likely to move to Ewood Park unless Rangers can step up their attempts quickly.
Source: PA
Source: PA