Thompson, who has also held senior marketing posts at Lastminute.com and Honda, joined Morrisons in 2011 from Apple, where he was a senior marketing boss.
Thompson is set to leave Morrisons at the end of the month, when his responsibilities will be taken on by marketing director Nick Collard.
Morrisons said Thompson would leave the company with "strong levels of customer satisfaction and operational metrics that are on plan".
News of his departure follows the announcement that CEO Philips is leaving at the end of March as Morrisons seeks new leadership to front a much-needed turnaround strategy.
The announcement of Philips’ departure came as the supermarket revealed a 3.1% fall in like-for-like Christmas sales for the six weeks to 4 January.
It was the first Christmas since Morrisons introduced its Match & More loyalty scheme, which it said had "launched well", although analysts have described it as "confusing".