The 29-year-old Spaniard, who is entering the final year of his contract at Anfield, was left out of the club's pre-season preparations along with Mario Balotelli and Fabio Borini.
Enrique, who lost his place in the matchday squad to teenager Joe Gomez for Liverpool's win against Stoke City in their season opener, has been linked with a move to West Bromwich Albion.
But the Spaniard says Liverpool's lack of experienced options at left-back could see him return to first-team action in the coming months.
“I'm in a difficult position,” Enrique said. “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,” he added.
“I will not give up. I know from experience that football changes from day to night.”
Enrique underwent surgery for a serious knee injury in 2013 and has failed to recapture his best form for the club ever since. He has made 66 Premier League appearances for Liverpool since joining from Newcastle United in 2011.
“I feel fine after two years with knee discomfort. Plus in the squad here we only have one pure left-back (Alberto Moreno) and I know that if I stay I'll play. I want to prove that some are wrong.”
Enrique said moving away from Anfield would be a step down in his career.
“What I think is that I will not play in a team better than this,” he said. “Liverpool are a match for Manchester United.
“I had a very good option to return to Spain, but 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.” – Reuters
Original source: Enrique vows to stay at Liverpool