The trio are surplus to requirements at Anfield and were left out of Liverpool's pre-season preparations.
Balotelli, Borini and Enrique were all left out of Rodgers' squad for the Reds' 1-0 win at Stoke on Sunday and have been training away from the first-team.
But Spanish left-back Enrique, 29, is determined to stay and fight for his place.
"I'm in a difficult position," he said. "The club have alienated Balotelli, Borini and me."
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Jose Enrique and Mario Balotelli are surplus to requirements at Liverpool
"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 knee surgery in 2013 and has struggled to recapture his best form since, but insists he is happy at Liverpool.
"I feel fine after two years with knee discomfort," the former Newcastle man added.
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"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.
"What I think is that I will not play in a team better than this. 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."