HOUSTON, Oct 29 (Reuters) - Exxon Mobil Corp. and Chevron Corp. on Friday disclosed plans to expand drilling in the top U.S. shale basin after posting their biggest quarterly profits in years. Latecomers to the west Texas shale fields, both last year slashed shale production and cut drilling as oil demand tanked. They could soon add two rigs each and rev up output, executives said on earnings calls. Exxon (XOM.N) last quarter produced about 500,000 barrels of oil and gas a day (bpd) from the Permian basin using 9 drilling rigs. "We may see a couple more rigs come on here," said Chief Executive Darren Woods, based on its latest well results. The company"s third quarter Permian output rose about 30% above the prior period, he said.Chevron (CVX.N) plans to add two drilling rigs and two crews to complete new wells in the Permian this quarter, Chief Financial Officer Pierre Breber told analysts. Its Permian production could rise to 1 million barrels per day from 600,000 bpd, Chevron said, on a day it reported $6.11 billion in net income, its best quarter since 2013. Exxon paid $30 billion for U.S. shale producer XTO Energy in 2010 in an ill-timed bet on natural gas prices and later spent up to $6.6 billion to add to its west Texas shale assets. In 2017, it took a $2 billion charge against its natural gas reserves from the XTO buyout, and last year wrote down the value of natural gas properties by about $20 billion, most associated with XTO, as demand for fuels tanked during the pandemic. Reporting by Sabrina Valle