Enrique says he wants to fight for his place in Brendan Rodgers' Liverpool side despite being ostracised from the first-team squad this summer.
The Spaniard, 29, has one season to run on his current Reds deal but does not appear to be part of Rodgers' thinking after the Northern Irishman left him out of the squad to face Stoke last Sunday.
Enrqiue, who joined the club from Newcastle in 2011, feels that he, along with Italian strike duo Mario Balotelli and Fabio Borini, have been "alienated" over the pre-season period.
However, after fully recovering from recent knee issues, the former Magpies full-back is not pushing for a way out at Anfield and is keen to provide competition for versatile teenager Joe Gomez and fellow countryman Alberto Moreno.
"I'm in a difficult position," Enrique said in quotes published in the Liverpool Echo.
"The club have alienated Balotelli, Borini and me.
"I have a year left on my contract and I imagine the club will want to sell, but I still want to stay and fight as I always have.
"But if a good offer came (from elsewhere) with more years of contract and more chance to play, I'd look at it.
"I will not give up, I know from experience that football changes from day to night.
"I feel fine after two years with knee discomfort. Plus in the squad here we only have one pure left-back and I know that if I stay, I'll play. I want to prove that some are wrong."
Enrique revealed the option to return to Spain, where he last played with Villarreal in 2007, had been on offer this summer.
However, despite not featuring for Liverpool since January, he sees any move away from Merseyside as a backwards one.
"What I think is that I will not play in a team better than this," he argued.
"Liverpool are a match for Manchester United. It is true we haven't won the league for many years but we must not forget that we have won five European Cups.
"I had a very good option to return to Spain but it is complicated because the economy is very different from what exists here.
"The only teams that can deal with what is paid in England are Real Madrid and Barcelona, so I would love to stay here until my legs stop me."