Greening was offered the welfare and pensions brief "but declined to take it," the source added.
"The prime minister is disappointed but respects her decision to leave the government."
Shortly after confirming Greening's departure, Downing Street officially announced she would be replaced by Damian Hinds, who had been serving as employment minister.
Greening, who represents a London constituency and backed Britain remaining in the EU during the Brexit referendum campaign, will return to the House of Commons backbenches.
Some Conservative colleagues immediately voiced their support for her.
Fellow member of parliament Heidi Allen wrote on Twitter she was "bitterly disappointed" at Greening's departure, saying: "A dreadful shame we have lost such a progressive, listening, compassionate woman from government."
May launched a long-awaited reshuffle of her Cabinet today, which nevertheless left most of her senior ministers in position.
Greening becomes the fourth minister since November to leave the government, after deputy prime minister Damian Green, defence secretary Michael Fallon and aid minister Priti Patel were forced out in unrelated scandals.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)