Tavernier has been linked with an £8 million move to Southampton, which was dismissed by Gerrard, pictured, while Morelos has been the subject of persistent speculation.
The Colombian striker scored his 21st goal of the season as Rangers won 3-0 at Livingston to move back into second place in the Premiership table, three points behind leaders Celtic who have a game in hand.
Gerrard is determined to hold onto both Tavernier and Morelos for the rest of the season but knows both could be the subject of offers before the January transfer window closes.
“I think there will be talk of Tav and talk of Alfredo but the reality is there hasn’t been a solid bid for any of our players so far,” said Gerrard.
“Will one come? Possibly, I’m not sure. I’ll have to answer the question when it does.
“There hasn’t been a respectable bid for any of the players where we stand right now. We made a call on it this morning and they [Southampton] are not [looking at Tavernier].”
Gerrard, below, was gratified by his team’s response to last Wednesday’s 2-1 defeat at Kilmarnock as Ryan Jack and Ryan Kent joined Morelos on the scoresheet at the Tony Macaroni Arena.
High profile signings Steven Davis and Jermain Defoe were restricted to substitute appearances as Gerrard made both tactical and personnel changes to the side which lost at Rugby Park.
“I thought we were excellent from start to finish today,”he added. “We matched Livingston in the fight and the battle early on and we competed with them in every single department and then our quality came through.
“The key today was we were perfect defensively. More often than not if we defend perfectly and we don’t gift people chances or goals, it gives us the chance to go and win games because of the quality we’ve got.
“If you’re comparing it with Kilmarnock, we beat ourselves on Wednesday night. It was nothing to do with tactics or formation it was to do with a defender gifting Livingston a lead and then a square pass and getting done on a counter attack, nothing to do with personnel or 4-4-2 or 4-3-3.
“The challenge for us now as a team is can we find consistency? Because we take one, two, three or four steps forward and then take a big step back.” If we want to become a good team and a successful team who are going to compete for what is available, we have to find consistency.”
Livingston manager Gary Holt had no complaints at the outcome on a day when his side never threatened to repeat the home victory they earned over Rangers back in September.
“We weren’t at the races,” said Holt, who confirmed Livingston’s interest in Queen of the South striker Lyndon Dykes. “Rangers did to us what we tried to do to them. We didn’t do the ugly side of the game well enough.”