Secretary of State George P. Shultz and Soviet Foreign Minister Andrei Gromyko met for 4 1/2 hours today, continuing discussions begun last week about arms control and adding an exchange of views on the Middle East, a senior State Department official said.

Arms control issues were a major topic when the two met for three hours last week, and the official said the United States "is neither encouraged nor discouraged by the Soviet position.

"It was the judgment of both that each is approaching the current talks in a serious and businesslike way," he said. "Obviously there is a lot of bargaining to go."

Discussions on intermediate-range and strategic nuclear weapons have provided the main points of contact for the United States and Soviet Union during the Reagan administration as relations otherwise have become strained, especially over the martial-law crackdown in Poland.

The two did not discuss a possible meeting between President Reagan and Soviet President Leonid I. Brezhnev, the official said.

One of several regional issues that came up, the official said, involved the Middle East, where U.S. diplomacy has pushed Moscow to the sidelines in recent years. "Obviously there are differences of view on the whole range of issues on the Middle East, but the discussion . . . was not inflammatory," he said.

The Soviet Union has been particularly critical of Reagan's Mideast peace initiative and has accused the United States of complicity in the recent massacre of Palestinians in two West Beirut refugee camps.

One point of agreement came on the Iran-Iraq war, which the two men were convinced "should end," according to the official. There reportedly were no specific suggestions about how to resolve the two-year conflict.

The meeting today at the Soviet mission to the United Nations, like the one last week at the U.S. mission, was characterized as "serious and businesslike, with no raised voices."

The fact that it went longer than had been expected raised the prospect of detailed discussions on some issues, particularly involving arms control. The official indicated that some arms-control issues discussed last week came up again.

U.N. discussions between the secretary of state and Gromyko have become a regular feature of U.S.-Soviet relations in recent years. The senior offical, who could not be identified under briefing rules established by the State Department, said the latest round was as much a "sizing up" session as an exchange of substantive views.

"This is the first time this secretary of state has met his counterpart. Both had chances for discussions in a good bit of detail. There was much familiar ground but some new ground on some subjects," the official noted.

Other regions and issues discussed included Afghanistan, southern Africa, Central America, human rights, the future of the European Security Conference and Soviet Jewry.