Citing "people familiar with the situation," the newspaper said the deal to make Kershaw the highest-paid pitcher in baseball history will be announced Friday.
The contract would well surpass the seven-year, $180 million deal signed by Detroit's Justin Verlander prior to last season.
At $30.7 million per year, it would also mark the highest annual salary ever awarded to any player.
ESPN reported the contract includes an opt-out clause after five years.
Kershaw, 25, is coming off a stellar season in which he claimed his second Cy Young Award in three years.
The left-hander had the lowest ERA in the major leagues at 1.83 and struck out a National League-best 232 en route to a 16-9 record over 33 starts for the NL West-champion Dodgers.
Over the past three seasons, Kershaw has gone 51-23 with 709 strikeouts in 697 innings.