BP is said to have granted 24-hour access to US government overseers and has pledged to meet safety requirements that go beyond tougher rules imposed after the accident, The Sunday Times reported.
A source close to the company said: "BP is hoping to resume drilling in the summer once it shows it can satisfy applicable regulatory conditions, as set out by the US offshore regulator."
A BP spokesman contacted would not comment on the story in a phone call to AFP.
BP will initially be allowed to only maintain or increase production on existing wells, ruling out exploration projects, The Sunday Times added.
The company's chairman Carl-Henric Svanberg last month said the 2010 Gulf of Mexico oil spill sparked by an explosion on a BP-leased platform was no reason to stop deep sea drilling.
"If we truly learn from this accident, I see no reason to close off the deep water as an area for future oil exploration and production," Svanberg told a conference on oil spill risk management.
Svanberg took over as chairman of the British group just a few months before the April 20 explosion on the BP-leased Deepwater Horizon platform in the Gulf of Mexico.
The incident killed 11 workers and sent some 4.9 million barrels of oil gushing into the Gulf over a three-month period, wreaking havoc on the region's environment and economy.
BP meanwhile faces having to pay compensation costs totalling tens of billions of dollars.
Following the spill, the largest ever manmade environmental disaster in the United States, Washington imposed a moratorium on deepwater drilling in the Gulf of Mexico, which it lifted in October.
The same month, a bid to freeze deepwater drilling in Europe in the wake of the Gulf of Mexico disaster collapsed under pressure from the multi-billion North Sea oil industry.