Mashaal asked for a comprehensive de-escalation that Israel and Hamas would start at the same time, involving an end to rocket attacks and to Israeli military operations in the Gaza Strip, the sources said.
Lavrov told Hamas political leader Khaled Mashaal during a meeting in Damascus Thursday that the group should stop its rocket attacks on Israel, the Arab newspaper al-Hayat quoted Palestinian sources as saying.
Lavrov accepted the proposal, according to the sources, as he believes any other option will not work.
Hamas also wants to include the opening of the Rafah crossing on the Egypt-Gaza border in the de-escalation proposals.
Hamas set conditions as a basis for any agreement on the opening of the border. The group rejects all forms of Israeli presence on the border, including electronic monitoring.
European border monitors should stay in Egypt's border towns not Israel, Hamas says.
The group rejects any presence of representatives of the illegitimate government of Palestinian Prime Minister Salam Fayyad but accepts representatives of forces loyal to the legitimate authority of President Mahmoud Abbas.
The group wants to have checkpoints manned by forces loyal to its Prime Minister Ismail Haniya on the Gaza-Rafah border and a share in revenues generated from custom duties levied on cross border trade.
Lavrov is on a Middle East tour to promote the idea of a peace conference that Moscow wants to host in May.