A run of eight points from their final nine matches saw the Reds drop to sixth, eight points off Champions League qualification - the target set by owners Fenway Sports Group at the beginning of the campaign.
Rodgers admits the shambolic performance at the Britannia Stadium, where his side found themselves five down at half-time, could be the final straw for FSG if they feel the need for change.
Asked if he understood the pressure was building on him the Northern Irishman said: "Yes, absolutely.
"I've always said that if the owners want me to go, then I go
It's as simple as that.
"But I still feel I've got a lot to offer the club
A lot has happened this year which has made the job difficult but I totally understand that.
"Last season, when things were working well, we had the support of everyone but performances like today do not help that and I fully understand that.
"For everybody connected with Liverpool, we are all embarrassed by that today.
"We have to be much better than that
They (the fans) deserve an apology.
"They were angry at half time - and rightly so - because that was awful, absolutely awful.
"The only way you'll get support from them is by winning games, and we haven't won enough games.
"They have every right to be angry and frustrated and of course I take full responsibility for that as the manager."
Rodgers also took responsibility for dropping Raheem Sterling for the first time in 18 matches and leaving him on the bench for the entire 90 minutes.
The 20-year-old has been splashed over the back pages all week after his contract wrangle with the club - having rejected a £100,000-a-week deal back in January - took more twists, with his agent reportedly saying he would not sign a new deal and the club responding by cancelling a scheduled meeting to reopen talks.
"With Raheem, there has been a lot going on over the past number of weeks and I felt there were other players mentally positioned to play in the game," was the manager's explanation.
In his two years in charge Stoke boss Mark Hughes has guided the club to back-to-back top-10 finishes in the Premier League for the first time.
"We couldn't have asked for a better finish to a really great season," he said.
"The level of performance in that first 45 minutes was exceptional and I don't think it reflects badly on Liverpool
I think most teams would have struggled to cope with the quality and intensity of our play.
"It is a great performance against a very good team with great individual talents, and we have been able to get the level of performance that I think a lot of people sensed we were capable of.
"Everything came together and we showed what a good side we are."
Stoke blitzed their hosts in a 23-minute spell from the middle of the first half, with Mame Diouf (two), Jon Walters, Charlie Adam and Steven Nzonzi putting them 5-0 up before the break.
Substitute Peter Crouch rounded off the scoring late on after Steven Gerrard, on his 710th and last appearance for the club, scored a goal which brought the whole of the Britannia Stadium to its feet.
"I am sure Steven wasn't anticipating the manner of that being his last performance, but credit to him he was trying to drive his team on and it was great he got that reception from our supporters when he scored," said Hughes.
"It is only right and proper we recognise the impact he has had in English football and at his club as well.
"To a man everyone applauded that goal
I wish him well, he has been a credit to his country and his club as well."
Source : PA
Source: PA