The midfielder made his debut for the Rams in Saturday's 2-1 home victory over Plymouth. He had to wait for the first appearance of his one-month loan while he completed a seven-match ban for receiving two red cards while playing for Spurs reserves at the end of last season.
The trip to East Midlands rivals Nottingham Forest at the weekend is due to be the final game of Livermore's stay at Pride Park but Clough wants to keep the 19-year-old for longer.
Clough, who felt Livermore improved as the game against Plymouth unfolded, said: "Jake was a bit tentative, like the rest of the team, early on but I thought once the first half-an-hour was out of the way he got into it.
"He gave the ball away for their goal but in the second half we saw a bit more of him making his runs forward, which is what he is in the team for - especially if he is going to play instead of Paul Green.
"It was his first competitive game for a long time and there is more to come from him and we are very pleased with him and with what we have seen of him in training."