The 36-year-old veteran of three World Cups, who has a drinking habit, missed Wednesday's first practice of the summer recess, turning up on Thursday but slinking away without speaking to reporters.
"It's true Ortega won't be staying at River," a club source who asked to remain anonymous told Reuters.
"They told me they would not be taking me into account," Ortega was later quoted as saying on the sports daily Ole's website when asked whether he would be part of the team for the championship starting next month.
"I've got to speak to (club president Daniel) Passarella to discuss rescinding my (contract)," added Ortega.
Local media said coach JJ Lopez had told Ortega he was not in his plans for the Clausura, the second of two championships in the 2010/11 season, due to the forward's "lack of commitment to the team".
Ortega, who played at the 1994, 1998 and 2002 World Cups, was in his fourth spell at River, the club where he made his first division debut in 1991 and with whom he has won five league and one South American titles.
His 1998 World Cup was marred by head-butting Netherlands goalkeeper Edwin van der Sar and being sent off in a quarter-final defeat.
He saw only limited action in the Apertura championship between August and December with weak performances and occasional absenteeism from training. River finished in fourth place 14 points behind winners Estudiantes.
A temperamental, skilled ball player nicknamed 'Burrito' (little donkey) with mesmerising dribbling and feints and brilliant goals from chips and set pieces, Ortega played for Valencia, Sampdoria and Parma during a not too successful spell in Europe from 1997 to 2000.
He left River again to join Fenerbahce in 2002 but did not stay long due to homesickness. He served a half-year suspension and was ordered by FIFA to pay a huge fine for breach of contract.
