The offer for paying debt almost certainly didn't happen, although it was related to another offer which definitely did.
It was made in 1984 by Escobar and Jorge Ochoa (another top member of the cartel). They wanted immunity in the courts in Colombia itself as well as protection from extradition. They promised to
Invest in Colombian industry, giving up their drug labs, airplanes, and air strips.
Promote crop substitution with farmers and work with social workers to stop the smoking of crack cocaine.
So more specifically, the promise was essentially to roll back all criminal operations in Colombia for the promise of state protection. The rumored part was the offer to also pay off the national debt, estimated between $9 and $10 billion at the time.
Could the cartel have paid the money? Maybe, but most likely not. We don't have an exact number for Pablo Escobar's wealth; according to Roberto Escobar, his brother and also accountant: “The exact number is impossible to know because so much of his money was involved with possessions whose value changed continuously". It clearly was in the billions; Forbes estimated him 7th richest in the world in 1989. Roberto talked about an incident where $7 million was hidden on a ship in refrigerators that ended up in Panama where the money disappeared. He quoted Pablo Escobar as saying
Son, what can we do? Sometimes we win, sometimes we lose.
That is, $7 million was a quantity so small it wasn't worth even blinking at. However, this doesn't mean $10 billion in liquid cash (even pulling from the cartel as a whole) was viable. A 1985 estimate from the Associated press put Pablo Escobar's wealth at $2 billion, later years estimated $3 billion. Some much less-well-sourced estimates go all the way to $25 billion in net worth for Pablo Escobar, but they seem to be based on fantasy, and for more realistic estimates a ~$10 billion bribe just doesn't seem feasible for 1984.
Additionally, and more fatally for the rumor, Jorge Ochoa very clearly stated in an interview with Frontline that paying the debt was never part of the offer.
We offered to the government that, in our respect, we would stop the business. But when they say that we offered to pay the debt, that was a total lie. That's a lie.
According to him, the part about stopping "business" in Colombia was the serious offer.
... I'm not quite sure why they said no after we had talked two or three times with the people sent from the government. They no longer talked to us ... But it would have been a great moment, and I think a lot of things would have been done, because after that a lot of violence happened. That would have been the best thing that could have happened with the government--for there to be peace ...
Given the level of violence in general Pablo Escobar was willing to escalate to -- allegedly thousands of deaths ordered directly by him in the prior year alone -- there perhaps was some skepticism the crime would really stop.
In 1986, there was yet another similar offer made to the United States. As reported by the Associated Press:
Leaders of the Medellin drug cartel in 1986 offered U.S. officials a deal under which they would halt drug trafficking and provide information on leftist guerrillas in Colombia in exchange for amnesty from prosecution...
Reagan turned the offer down flat, with the assistant secretary of state for international narcotics stating "We don't do business with international outlaws."
...
Escobar, R. (2009). The Accountant's Story: Inside the Violent World of the Medellín Cartel. Grand Central Publishing.
Hinterseer, K. (2002). Criminal Finance: The Political Economy of Money Laundering in a Comparative Legal Context. Springer.