The 37-year-old had indicated the second Test in Wellington could be his last with the intention of retiring from all international cricket after the World Cup which starts next month.
But after a match-changing innings of 203 to lead Sri Lanka out of trouble and put them in control of the Test at the Basin Reserve, Sangakkara said the lure of overtaking Bradman meant he may extend his career.
"I would love to," he said, reflecting on the impact his seven-hour innings had on a match where he engineered Sri Lanka's recovery from 78 for five to 356 all out and a first innings lead of 135.
"I've had a chat with the (Sri Lanka) selectors and promised them I will really have a chat and reconsider to see whether there is a few more months of cricket in me Test-wise.
Also Read
Sangakkara, the most prolific Test batsman in the world last year with 1,493 runs, began this year by becoming the fifth player, and the fastest, to reach 12,000 Test runs.
Of his 200-plus innings, five have been away from home to equal the record held by Bradman, Len Hutton and Brian Lara.
He is now fourth on the list of century-makers with 38, behind India's Sachin Tendulkar (51), South Africa's Jacques Kallis (45) and Australia's Ricky Ponting (41).
Among the top echelon of Test batsmen, Sangakkara's average of 58.92 is second only to Bradman.
"The desire is always there," Sangakkara said as he weighed up the statistics against thoughts of retirement and spending more time with family and business interests.
Should Sangakkara extend his career, Sri Lanka have Test series coming up this year against Pakistan, India, the West Indies and New Zealand again.