Official results of Sunday's vote are expected on Thursday, but both Sall's ruling party, Benno Bokk Yakaar (BBY), and the opposition have claimed victory, citing partial results.
"Macky Sall wants to confiscate our victory but we will not accept it," opposition leader Ousmane Sonko of the coalition Yewwi Askane Wi (YAW) said at a news conference.
"I am asking the Senegalese people to defend their votes by all means."
In response, Cabinet minister and ruling party member Abdoulaye Seydou Sow said "these threats are unacceptable."
Tensions have run high in the politically stable West African country of 17.5 million since violent protests broke out last year, when Sonko, who came third in the last 2019 presidential election, was arrested on rape charges. Sonko denies the allegations.
On Monday, another opposition coalition known as Wallu Senegal said it and allied YAW had defeated the ruling party in most departments. The ruling coalition said it had won 30 out of Senegal's 46 administrative departments.
At Wednesday's news conference, YAW representative Dethie Fall said YAW would not accept Thursday's results. YAW made allegations that there were voting irregularities at polling stations in northern regions.
YAW plans to ask the appeals court on Thursday to open ballot boxes, so YAW can show evidence to back up its allegations, said Fall.
The ruling party's Sow later said YAW's claims were baseless.