Almeyda, who has made a remarkable comeback after four years of five-a-side football and lower division drudgery, suffered a thigh injury in the 1-1 draw at home to Quilmes on Sunday and could be out for at least four weeks.
The 36-year-old former Argentina midfielder, who played at the 1998 and 2002 World Cups, has been instrumental in River's improvement this season and his absence will give Cappa a tactical headache.
River hope Almeyda, who is open-minded about how much longer he can last in top-flight football, can come back for the match away to All Boys at the end of October. River are at home to arch-enemies Boca in Argentina's "superclasico" the following weekend.
"I'm not broken yet, I'm just on one leg," Almeyda told a news conference at River's Monumental stadium after Tuesday's training. "My legs will have to be cut off for me to quit.
"I'm not basing my hopes specifically on the match against Boca. I want to play all the matches."
Almeyda, a Libertadores Cup winner with River Plate in 1996, reached a pinnacle in his career by helping Lazio win the Italian Serie A championship in 2000.
Without actually announcing his retirement, he appeared to have virtually walked out on the game in the late 2000s only to return to River last year.
River, fifth with 14 points three behind leaders Velez Sarsfield after eight matches, are away to sixth-placed Banfield, Apertura winners last season, this Sunday.
Boca are also hoping to have Juan Roman Riquelme back dictating their game in midfield by the time of the match against River, a derby with a potential to save a disappointing season in the eyes of fans.
Riquelme is recovering from knee surgery and has yet to play this season while Boca have been unconvincing and erratic under new coach Claudio Borghi.
Boca, in 12th place with 10 points after losing 1-0 to Estudiantes last weekend, are at home to Lanus on Sunday.
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