June 1 (Reuters) - Most Gulf stock markets fell at the start of trading on Monday after an exchange of fire between Iran and the United States, as Israel instructed its forces to push further into Lebanon as part of its battle with the Tehran-backed Hezbollah group.

The United States said it struck Iranian military sites earlier this week, and Iran’s Revolutionary Guard said on Monday it targeted a U.S. base in retaliation, in the latest exchange of blows despite negotiations aimed at ending the war that began three months ago.

The escalation, which came after peace talks between Israel and Lebanon hosted by Washington on Friday, has diminished hopes that the United States and Iran will soon announce an agreement to ensure the continuation of their ceasefire.

Saudi Arabia’s main index fell 0.1 percent, weighed down by a 0.5 percent drop in Al Rajhi Bank shares.

Qatar's main index fell by more than one percent amid declines in most of the listed stocks, including Qatar Islamic Bank, whose shares dropped 2.2 percent.

In Abu Dhabi, the main index fell 0.1 percent, pressured by a 2 percent drop in shares of the country's largest bank, First Abu Dhabi Bank, amid exposure to defaults linked to the collapsed mortgage finance company Market Financial Solutions, according to the Financial Times.

The bank did not immediately respond to a Reuters request for comment.

Dubai’s main index bucked the trend and rose 1.1 percent, supported by a 3.7 percent increase in shares of Salik, the operator of the toll road system.

Trump said on Friday that he would soon make a decision on a proposed agreement to continue the ceasefire with Iran, although it remains clear that disagreements persist between the two sides over key issues that have been central to the conflict.

Oil prices rose more than 2 percent in early trading on Monday, raising concerns about inflation and rising interest rates.