US President Barack Obama is at loggerheads with his Pakistani counterpart Asif Ali Zardari, refusing to meet him apparently over blockage of NATO supply lines into Afghanistan, an issue which continues to strain ties between two nations, American media reports said on Monday.

The New York Times said a deal to reopen the supply lines fell apart as Obama began talks on ending the NATO alliance's combat role in Afghanistan in 2013.

As a two-day NATO summit meeting opened in Chicago, Obama remained at loggerheads with Zardari, refusing even to meet him without a deal on the supply routes, which officials in both sides acknowledged would not be coming soon, it said.

Zardari, who flew to Chicago with hopes of lifting his stature with a meeting with Obama, was preparing to leave empty-handed as the two countries continued to feel the repercussions of a fatal American air strike last November, for which the US President has offered condolences but no apology, the paper said.

But White House Deputy National Security Advisor Ben Rhodes said that President Obama could not meet Zardari as he had a 'full slate of summit meetings to attend'.

"The two bilateral meetings, really, that he did, or President (Hamid) Karzai for obvious reasons given the focus on Afghanistan here, and the secretary general of NATO given that it’s traditional for the host to make sure that we're aligned with the secretary general heading into the summit.''

"But we don't anticipate any other bilateral meetings so we didn't draw that linkage. We're going to continue to work through the issue with the Pakistanis," he said. Zardari did, however, meet with Secretary of State Hillary Clinton to discuss the supply routes.

A State Department official later said that Clinton and Zardari discussed the importance of reopening NATO supply lines and of taking joint action against the terror groups, in Af-Pak region including al-Qaeda and Haqqani.

"This whole breakdown in the relationship between the US and Pakistan has come down to a fixation of this apology issue," Vali Nasr, a former State Department adviser on Pakistan, was quoted as saying by the daily.

The combination of no apology and no meeting, Nasr said, 'will send a powerfully humiliating message back to Pakistan'.

According to the daily, the failure to strike a deal on the supply routes ahead of the summit injects new tension into the relationship.

"When NATO extended the invitation, we thought it would move the Pakistanis off the dime," a senior American official was quoted as saying.

Without the deal, "it's going to be really uncomfortable" for Zardari at the summit, which runs through Monday, the official told The New York Times.

The Washington Post said that hopes that the deal on reopening of the supply routes would be concluded before the summit were not realised as the two sides continue to haggle over new tariff rates Pakistan wants to impose.

After weeks of closed-door negotiations with Zardari's government, US officials did not deny that they are seeking to send the Pakistanis a public message, The Los Angeles Times said.

"If they're feeling a little bit of pressure this weekend, they should," a US official was quoted as saying.