Thousands of supporters of the center-right opposition attended gatherings in Caracas and across the country seeking a referendum on cutting his term short.
But launching a referendum is the role of the national electoral board, which the opposition accuses of being controlled by the president's allies.
The opposition vowed to oust Maduro when it took control of the legislature in January after winning elections, blaming him for a crippling economic crisis.
Seeking to open another front, the opposition also plans to unveil a bill in the legislature this week to cut short Maduro's term.
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He has vowed to hold on to power and press on with the socialist "revolution" launched by his late predecessor Hugo Chavez.
Addressing thousands of supporters who gathered in heavy rain yesterday outside the presidential palace in downtown Caracas, Maduro focused most of his attacks on the opposition-controlled Parliament.
The efforts are "the spearhead of a coup that aims to once again promote North American imperialism against the revolution," he said, as Venezuela observed a holiday marking its struggle for independence.
Observers and politicians in the South American country have warned that economic hardship and political tensions could spark violence.
"This country is a bomb that could explode," leading opposition figure Enrique Capriles said at one of the anti-Maduro gatherings.
Venezuela's economy has plunged along with the price of the oil on which it relies for foreign revenues. Shortages of medicines and goods such as toilet paper and cooking oil are widespread.
Maduro blames the collapse on an "economic war" by capitalists. The opposition blames his economic management.
"We have had three hard years of struggle against the economic war. We have come to tell the president that the people are with him.