Cole made a big impact on Ligue 1 after spending the season with former champions Lille, leading to speculation that the French outfit would be keen on taking him on for another 12 months.
An emotional return to Upton Park has also been mooted, where he emerged through the youth ranks before spending five years as a senior player.
However, the 30-year-old midfielder has not given up hope of resurrecting his career on Merseyside, with the club currently on the lookout for a replacement for former manager Kenny Dalglish.
"When I signed for Liverpool, it startled me a bit," Cole said as reported by the Daily Mail.
"I don't know if I felt the pressure but the city is so in love with football. It's a religion. If I go back, I will be more prepared for it this time.
"I can see why people say [joining Liverpool] is a culture shock. It is a small place but it has this great buzz. How can there not be in a city that has produced five Prime Ministers and The Beatles? The people are so sharp, quite a bit like Londoners really.
"There is still a part of me that would really love to make it work there and I don't want to have a bad spell at Anfield on my CV. But it is out of my hands now.
"At the minute I can't rule anything in or out. We have got to speak to Liverpool and see what happens there."
‘Slot hasn’t come in and changed everything, making it all about himself – it’s been about the players he’s inherited, getting the best out of a system he likes’: Liverpool manager impressing former players with personality and tactical approach
‘We thought we’d signed him, but Chelsea blew us away. I think they signed him so we couldn’t have him, as much as because they really wanted him’: Liverpool actually failed to sign Mohamed Salah in 2014, three years before finally getting the Egyptian
