Smith is attracting interest after scoring 12 times as an ever-present in the Hornets' Championship campaign this term.
But Rodgers wants to keep him at the club to help their relegation fight after his side dropped into the bottom three at the weekend.
He told the Watford Observer: "There was an enquiry but certainly, as far as I am aware, there wasn't an official bid. But certainly there was an enquiry around the mark mentioned (£1.6 million).
"I will be happy when Tuesday comes. This is my first transfer window and there has been a lot of speculation and other things in the last couple of days.
"Tommy was excellent on Saturday, he led the team very well and had another good performance and I will be glad when the third (of February) comes and we can see it through and hopefully he will be here until the end of the season."
Rodgers added: "I think his worth is greater than that (£1.6 million). Obviously the club and the board have to look at what the offer is and depending on the state of the club and where the club is financially, then they will decided whether it is acceptable or not.
"But what I know is that the value and worth of Tommy is certainly more than £1.6 million when you look at the fees of other players going around."
Rodgers confirmed that Cardiff's approach was the first one Watford had received, while the player himself wants to stay at the club.
He said: "I am confident that he (Smith) will be here for the remainder of the season.
"He wants to be here, we have had a chat and it would need to be something very good, for him and the club, if it was to take him away from the club and at the moment, I am totally happy with where we are at.
"It has been quite quiet to be honest. Obviously he is a big player for us and clubs are assessing and looking at the situation but certainly for him it is the first one (enquiry) we have had."