The Reds haven't won in normal time since back on August 17 when they squeezed past Bournemouth 1-0, but eventually saw off the threat of League Two side Carlisle United on penalties in the Capital One Cup on Tuesday.
The first half away to Arsenal aside, Rodgers' side have failed to perform all season, and seven points from their first three games represented a solid start for the Champions League hopefuls, but perhaps masked the fact that the Reds had failed to find their second gear.
However, back-to-back defeats to West Ham and Manchester United either side of the international break and Sunday's 1-1 draw with Norwich City has cranked up the pressure on Rodgers, with fans unhappy with the unattractive brand of football being served up at Anfield so far this season.
Former Borussia Dortmund boss Jurgen Klopp is available, despite signalling an intent to take a sabbatical after leaving BVB at the end of last season, while the legendary Carlo Ancelotti has also been linked with Rodgers' job.
Speaking to Channel 5, Hutchison, who played for Liverpool from 1990-1994, feels that Klopp's availability is a bad omen for Rodgers.
Hutchison said: “You do worry for Brendan and when you see Jurgen Klopp and Carlo Ancelotti being linked it is worrying. It was not a good performance [against Carlisle].
“I don’t think it is enough [to calm fans]. I think it weakens his position, and they never got a good result against Norwich at the weekend and tonight they should have had enough to get through."
Rodgers' men are now six games without a win (penalty shootout victories over League Two clubs aside) and they return to action this weekend when they host Aston Villa.
Rodgers made six changes to his team for the visit from Carlisle, but the likes of Daniel Sturridge, Philippe Coutinho and Lucas Leiva could be recalled for the visit of Villa, while Mamadou Sakho will replace Dejan Lovren who has a suspected broken ankle.
However, Liverpool will host the Villans without their £32m striker Christian Benteke, with the ex-Villa forward ruled out with a hamstring problem, and the Belgium international faces a race against time to be fit in time for next Sunday's Merseyside derby.