George P. Shultz began conferring with members of Congress yesterday, and Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) predicted that the Senate will confirm him as secretary of state "after a searching inquiry into his attitude toward U.S. foreign policy in the Middle East."

Shultz's position as president of the Bechtel Group Inc., a conglomerate with major interests in Saudi Arabia, and his past advocacy of a more evenhanded U.S. approach to the Arab-Israeli conflict have made the Middle East the potential focus of the upcoming Senate hearings on confirming him as successor to Alexander M. Haig Jr.

While saying that they expect Shultz to be confirmed, several pro-Israeli senators have indicated that they intend to probe closely into his views on Mideast policy during the Senate Foreign Relations Committee hearings.

Committee sources said yesterday that the hearings are expected to begin July 13. They added that Chairman Charles H. Percy (R-Ill.) hopes to complete them that same day but conceded that involved questioning about the Middle East might require additional time.

Sen. Alan Cranston (D-Calif.) said after meeting with Shultz yesterday that it would be "presumptuous" to make a pro or con judgment on the nominee at this point. Cranston added, however, that he and other committee members will use the hearings as a forum to determine whether President Reagan's choice of Shultz "means a tilt toward the Arab side or continued support of a policy that recognizes Israel as the only democracy in the Middle East."

Cranston, noting that Defense Secretary Caspar W. Weinberger also came into the administration from Bechtel, said he was concerned about whether it is wise to have "one corporation serve as the base for both the secretary of defense and the secretary of state."

He continued: "I am concerned at the fact of Bechtel's very extensive business with the Arab world and the fact that it has no business with Israel and what that means about the judgments Mr. Shultz might have made in the past or, more importantly, will make in the future."

Another strongly pro-Israeli senator, Rudy Boschwitz (R-Minn.), chairman of the Mideast subcommittee, said he had been impressed by Shultz's "very broad range of knowledge" about the Middle East. In response to questions about Shultz's business background, he said: "In the balance of things, it has to be troublesome . . . . It's a concern that always exists, particularly when you're dealing with the State Department. But it's too early to say."

In addition to meeting with Baker and various Foreign Relations Committee members, Shultz paid calls on Senate Minority Leader Robert C. Byrd (D-W.Va.), House Minority Leader Robert H. Michel (R-Ill.) and Rep. William S. Broomfield (R-Mich.), ranking minority member of the House Foreign Affairs Committee. His consultations on Capitol Hill are to continue today and Thursday.

Most senators who met Shultz yesterday praised him personally but said they would reserve judgment on voting to confirm him until after the hearings. However, Baker said: "I expect that he will do very well in his hearings and will be confirmed by the Senate. I don't expect him to have any serious problems."

At the State Department, speculation continued about whether Haig, who has remained in charge temporarily to oversee management of the Lebanon crisis, will fade away quietly after Thursday evening, when he begins a long holiday weekend. At the White House, a spokesman said only that Haig's formal departure time "is still to be worked out."

Department sources said that Lawrence S. Eagleburger, undersecretary for political affairs, had been asked to remain in the post and had agreed to do so. Eagleburger is the third-ranking official in the department and his job involves supervising the day-to-day management of most major policy issues.