The Andalusians have not won in La Liga since early November and have slipped to 16th out of 20 in the standings, one point above the relegation places.
They were also humiliated by second-division Hercules in the King's Cup, losing home and away to exit 3-1 on aggregate.
"I respect the opinion of the fans and I understand that people will protest," Sanchez, who was appointed just under a year ago to replace the sacked Gonzalo Arconada, told a news conference.
"We have to have confidence and patience because the team is playing good football and the goals and victories will come soon if we carry on in this vein.
"I would be worried if the team was playing badly but that's not how it is. What we need is support and encouragement."
Widely regarded as the finest player Mexico has produced, the former Real Madrid striker became coach of the national side after the 2006 World Cup but was sacked in March 2008.
The 51-year-old's position at Almeria appeared precarious on Sunday after club president Alfonso Garcia Gabarron was quoted as saying in the Spanish media he could not give assurances Sanchez would remain in charge.
"I have full confidence in my team because the players are working and playing well and I am convinced we will manage to stay in the first division," Sanchez added.
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