United fielded several enquiries about the 21-year-old striker earlier in the week and Rangers were reported to have made an offer on Friday as they prepare for the potential loss of Birmingham target Kenny Miller.
But Houston, whose side started the defence of their trophy slowly before seeing some late pressure repelled by last season's beaten finalists, said: "No formal offer has been made."
He added: "I'm not expecting any because the chairman hasn't said that anyone else has been on. If David Goodwillie goes and we get the right money for him, good luck to him. I will be the first to shake his hand because, the way he is performing, he has earned maybe getting a move to a bigger club."
United struggled to create chances in the first half and Houston's dissatisfaction with his strikeforce was evident when he replaced Jon Daly and Johnny Russell with Andis Shala and Danny Cadamarteri in the 58th minute.
Houston felt his side got "slightly better" after the change and County survived some late pressure with Morgaro Gomis drawing a good save from Michael McGovern before Scott Robertson and Shala were denied by impressive clearances from Stuart Kettlewell and Scott Morrison.
Goodwillie has scored 11 of United's 21 goals this season, but they were also without the likes of Craig Conway and Danny Swanson through injury and Houston is looking for all-round improvement.
"Let's be honest, with the quality of striker he is, yes we missed him," Houston said. "It would be foolish to say otherwise. He is on a hot streak of form, scoring goals every week, and take him out the team and we are not scoring enough goals.
"We didn't create an awful lot, especially in the first half. But we have not been here playing for a couple of months and I think it showed. We need to get sharper all over the pitch.
"There were aspects of the game that were pretty poor from us, and we have a lot to work on. I didn't think we were in any great danger of going out of the cup, the better chances fell to us, especially in the second half, although we were nowhere near our best."