Everton, Tottenham, Aston Villa, Swansea and Bournemouth all had representatives at Anfield to see him in action on Thursday evening.
They will have been impressed with what they saw as well, as Saivet converted a superb free kick to hand his side the lead in the first half.
The Senegal international has been converted into a central midfielder recently, having played further forward as a winger or a forward before then.
He is available for as little as £3.5miliion in the January transfer window, according to the Mirror, and that has led to the widespread interest in his services.
Of that bunch of clubs, Everton are thought to be the most long-term admirers, having watched him before now.
Sunderland and Southampton have also shown interest in him in recent seasons, but did not have anyone watching him this time around.
Whether any side would fancy a move in January remains to be seen, but for such a low price, many would surely consider it worth the risk.