Thaksin Shinawatra, who was ousted as prime minister in a 2006 military coup and now exiled, had called on his supporters, whom many people link to the red shirts, to seek "reconciliation" following the offer.
Al Jazeera's Wayne Hay, reporting from Bangkok, said Abhisit had said the offer had been the last chance for the red shirts to reach an agreement, and had set Tuesday as a firm deadline for the protest leaders to respond.
Pressure has been building on the red shirts to end their two-month protests which have triggered clashes with government forces that have claimed 27 lives and left about 1,000 injured. Thousands of red shirts remained in the protest site on Tuesday.
Businesses in the capital's main commercial area and the crucial tourism sector have been badly affected and a red-shirts raid on a nearby hospital last week looking for soldiers was roundly criticised.
Our correspondent said that 24 red shirt leaders still face arrest - some for criminal offences - and one of their conditions is amnesty for those involved in the protests.
Abhisit has said he is willing to consider amnesty for those who violated emergency regulations, but not for those who committed criminal offences.
Abhisit rejected an offer last month by the red shirts to end their siege of Bangkok's business district in return for elections within three months.
He also previously offered an election in December, about a year before his term ends.
Analysts have said that both sides want to be in power in September for two critical events: a reshuffle of leaders in the military and police force, and the passing of the national budget.
In his televised statement on Monday, Abhisit set five broad conditions for reconciliation.
The monarchy, he said, should not be dragged into politics or "violated" - a condition that follows government accusations some red shirts aim to overthrow the monarchy.
The second condition calls for reforms to address social injustice, the third proposes an independent body to monitor media to ensure unbiased reporting, and the fourth a committee to investigate recent political violence.
The fifth point broaches possible political reforms that could include constitutional amendments and a review of a five-year ban on politicians allied with Thaksin.