Tony Hayward, the beleaguered chief executive of BP, has claimed its oil spill in the Gulf of Mexico is "relatively tiny" compared with the "very big ocean".

In an bullish interview with the Guardian at BP's crisis centre in Houston, Hayward insisted that the leaked oil and the estimated 400,000 gallons of dispersant that BP has pumped into the sea to try to tackle the slick should be put in context.

"The Gulf of Mexico is a very big ocean. The amount of volume of oil and dispersant we are putting into it is tiny in relation to the total water volume," he said.

US authorities estimate that about 5,000 barrels of oil a day are leaking from the damaged pipeline lying on the seabed after attempts to stem the flow failed.

The spill began just over three weeks ago when a buildup of gas erupted from a well being drilled by BP in seabed about 1,500 metres (5,000 ft) below sea level. When the gas ignited at the surface, the explosion sunk the rig, the Deepwater Horizon, and 11 workers were killed.

Hayward promised that BP would "fix" the disaster, which is on course to eclipse the 1989 Exxon Valdez oil spill as the biggest US oil spill in history. "We will fix it. I guarantee it. The only question is we do not know when."

But the BP boss admitted for the first time that his job was on the line because of the Gulf of Mexico disaster. Asked if he felt his job was already under threat, he replied: "I don't at the moment. That of course may change. I will be judged by the nature of the response."

Hayward stressed that BP's efforts to contain the spill had succeeded in dispersing the oil and preventing large amounts reaching the shoreline around the Gulf.

But environmentalists are concerned about the unseen damage being done to marine life by the oil, which is sinking to the seabed.

Hayward said that BP was "increasingly confident" that progress was being made in trying to find ways to seal the faulty blow-out preventer, which failed to shut down the well and stem the flow of oil.

This week, US politicians attacked BP and the other companies involved in the drilling when they gave evidence at Senate hearings into the disaster.

But Hayward insisted that deep-water drilling would continue in the US despite the growing environmental and political backlash against the company.

"Apollo 13 [the unsuccessful third mission to the moon in 1970] did not stop the space race," he said. "Neither did the Air France plane last year coming out of Brazil [which mysteriously crashed] stop the world airline industry flying people around the world. It's the same for the oil industry."

He pointed out that the Gulf of Mexico, much of it in deep waters, represented one-third of the US's oil and gas production.

He insisted that BP was in far better shape to respond to the disaster than it was five years ago under his predecessor, Lord Browne. In 2005, 15 workers died in an explosion at BP's Texas refinery.

BP was then fined a record £53m by the US authorities, who also criticised the company's safety culture. BP also suffered a pipeline leak in Alaska in 2006.

"In the last four or five years we have made major improvements in safety performance. It has made the company much better … Four years ago it could have been very different," Hayward said.

BP's share price has slumped since the disaster. BP told the City yesterday it had now spent $450m (£304m) fighting the oil leak, but some analysts put BP's total bill for the clean-up and damages at $23bn. President Barack Obama has now proposed scrapping a cap on BP's liability for the spill.

Transocean, the world's largest offshore drilling contractor which owned the Deepwater Horizon, filed papers in a Houston court on Thursday seeking to limit its legal liability to $27m.

Hayward said it was "unwise" to speculate about the direct causes of the accident before investigations had been completed. "There is a lot of speculation, red herrings and hearsay." He also admitted that BP had made mistakes in its early response to the crisis. It initially refused to compensate fishermen who were unable to produce written proof of their normal earnings. Most keep no such records.

He also said BP had made a mistake when fishermen signing up to help with the relief effort were required to sign agreements limiting their receipt of any future damages from BP.

"It was a bit bumpy to get it going. We made a few little mistakes early on."