Hodgson took charge of Liverpool under difficult circumstances this summer, having to work with limited transfer funds after last season's seventh-place finish.
However, results so far have still been disappointing, with the club sitting 16th in the table after five games, and Wednesday's shock cup exit to Northampton, who are currently 17th in League Two, added extra pressure.
Bookmakers have subsequently slashed odds on Hodgson leaving the club, but the current board have not discussed his future and the matter has not arisen during takeover talks.
A club insider told ESPNsoccernet: "I don't believe that the manager's position is unduly under threat. After all, last season Roy Hodgson was manager of the year and he is only five games into the new season.
"The big problem is that the manager no longer has the resources to fund a competitive team. That is the harsh reality.
"As for takeover talk, well, the identity of the manager, whether the existing one or a new one, has never cropped up in all the talks as far as I know. It is not an issue in the sale process and the subject has not been approached.
"Let's face it, of course it's the prerogative of a new owner to change the manager - that is always a possibility - but there are no bids on the table, and the club is some way off getting a new owner, if it gets one at all.
"So, for the time being the issue of Roy Hodgson is purely hypothetical and speculative."
Liverpool captain Steven Gerrard, meanwhile, is confident the club will bounce back from this latest setback.
"Football isn't just about highs," he said. "Throughout my career I have experienced both highs and lows and probably learnt more from the lows because you look at yourself first, look at what went wrong and try to learn from your mistakes.
"The players are allowed to be down today but it's up to players like myself to pick them back up. Playing for this club there are big games every two or three days and we need to get everyone's head focused on Sunderland now.
"What I would ask all the supporters is to trust the players and what the players are saying. I've been reading and listening to what they've been saying and they've all said we'll get it right in the end - and we will."