Rangers are within five points of Celtic in the Scottish Premiership title race with two games in hand after an impressive run under Clement.
They have also won the Scottish League Cup and reached the last 16 of the Europa League as group winners.
Clement has been insistent that he is not content with what his team have achieved already and recently mentioned winning more trophies in Scottish football going forward.
Hartson feels the Rangers boss has laid down the gauntlet and now Celtic and Rodgers must react to the Belgian.
The former Bhoy insisted that the Celtic manager must vow to make sure that Clement does not achieve his goals.
Hartson said on the Go Radio Football Show: “When Clement is talking about titles, maybe he is saying not this season, but if that is not a warning sign for Celtic when the opposition manager is pointing out he wants to win titles.
“You have got to respond to that.
“You have got to react to that comment and say while I am manager here I am going to make sure that I got to the board, I am going to bring in the right players to suit the club and the way I want to play and I am going to stop you from winning titles.
“That’s the way Brendan Rodgers would be thinking.”
Celtic are expected to spend money in the winter window to add more quality to their squad.
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