President Reagan announced yesterday that he will nominate Army Gen. John W. Vessey Jr., a 59-year-old former "mud soldier," to become chairman of the Joint Chiefs of Staff, effective July 1.

Assuming Senate confirmation, Vessey will succeed Air Force Gen. David C. Jones, who was known in military circles more as a politically adept manager than as a down-to-earth soldier.

Vessey, according to the Army, would be the first Army general to become chairman of the Joint Chiefs without having graduated from the U.S. Military Academy at West Point.

His selection also is unusual in that he would jump from his current job as Army vice chief of staff to the chairmanship without first becoming the chief of staff of his service.

The chairman of the Joint Chiefs runs the meetings of the heads of the Army, Air Force, Navy and Marine Corps and sets forth their positions in meetings with the president. He is the president's military adviser on the main issues of the day.

A White House official said that President Reagan did not know Vessey but directed his staff to find a man "of absolute integrity who would be cool under fire." Reagan was particularly impressed, the official said, with the fact that Vessey was a former enlisted man who received a battlefield commission on the Anzio beachhead during World War II.

"A soldier's soldier," Reagan said of Vessey as he announced his choice to reporters yesterday just before boarding a helicopter at the airport in Santa Monica, Calif., en route to his nearby ranch for a vacation.

Army Secretary John O. Marsh Jr. praised Vessey's "down-to-earth, wise counsel" as that service's vice chief of staff.

Vessey's current boss, Gen. E. C. Meyer, Army chief of staff, also hailed his selection, declaring: "I can't think of a finer leader who can articulate the security needs of our country."

The initial reactions of the Army's old-boy network were favorable as well.

Typical descriptions of Vessey were "the best of the four stars"; "wise old man"; "cautious and conservative"; "quiet, thoughtful"; "adds some balance because he knows the ground war during this pursuit of navalism," and "has been down the road a lot further than the other chiefs."

Although there was speculation that one of the current members of the Joint Chiefs would be nominated to become chairman--with Adm. Thomas B. Hayward, chief of naval operations, and Marine Corps Commandant Robert B. Barrow frequently mentioned--Reagan seemingly wanted to start fresh with a senior soldier.

"Age is a little more in nowadays," quipped one Army officer.

The current chiefs are stained by their planning and approval of the 1980 attempt to rescue the American hostages in Tehran, a mission that ended in flames in the Iranian back country. But a White House official said this did not figure in Reagan's selection.

A native of Minneapolis, Vessey started his military career by enlisting in the Minnesota National Guard in May, 1939. He was called to active duty in 1941, rose to the rank of sergeant, and was commissioned a second lieutenant in the field artillery at the Anzio beachhead on May 6, 1944.

His early service was in ground combat divisions, including the 34th Infantry Division in North Africa and Italy, the 4th Infantry and 3d Armored Division in Germany and the 25th Infantry Division in Vietnam.

Promoted to full general in 1976, Vessey became commander of U.S. forces in South Korea that year, returning to the United States as Army vice chief of staff in 1979.

He is married to the former Avis C. Funk, and the couple has two sons and a daughter.