The Class A baseball team scheduled to be born in Alexandria in April already is embroiled in controversy -- from within and without.

A search for 20 players to stock the Carolina League team has started. But the club owners and league officials disagree on the caliber of players to be sought and where they will come from.

And Alexandria property owners have gone to court in an attempt to halt the City Council from spending money to improve the grounds where the team will play.

Eugene Thomas, a local builder who is president of the Alexandria Baseball Club, Inc., the expansion team's owners, said yesterday he will hire only those players that have been signed by major-league teams but have not yet been placed in the minors. "I can tell you this. there won't be anybody here that's been cut from the major leagues," said Thomas.

However, James B. Mills, president of the Carolina League, and Les Peden, the new team's manager, said they expected the Alexandria club to hire players who could not make the major-league farm clubs.

"I'm sure they will be cutting a lot of Double A ballplayers that don't have any major-league prospects," said Peden, who has managed minor- league teams for 12 years and was a catcher in nine games for the Washington Senators during the 1953 seasons. Team General Manager Michael Halbrooks could not be reached for comment.

Even as the owners raced to put a team together by the April 14 opening game, the political controversy surrounding the decision of the Alexandria City Council to appropriate $120,000 for field, street and parking improvements at the Cora Kelly Elementary School continued.

George Pope, president of the Lynhaven Citizens Association, has filed a petition in Alexandria Circuit Court seeking an injunction to enjoin the city from spending the money, on grounds that the project was not included in the city's 1977-78 capital improvement budget as is mandated by the city charter. The petition also charges that the project was not approved by the city planning commission. A hearing has been set for March 8.

A spokesman for Baseball Commissioner Bowie Kuhn said yesterday that the commissioner does not intend to block the team from locating in Alexandria.

The spokesman said Kuhn has assurances from the Alexandria club, the Carolina League, the city's mayor, and minor-leagues president Bobby Bragan that they would not fight a major-league club's entry to Washington, or seek indemnification.

The Carolina League, considered with the California League as the best Class A league in baseball, had four teams in Virginia and North Carolina last year. In addition to Alexandria, an expansion franchise is being formed in Kinston, N.C.

All four current teams in the league are associated with a major-league club, with the Peninsula team in the Tidewater area being owned outright by the Philadelphia Phillies. The other three -- the Lynchburg Mets, Salem Pirates and Winston Salem Red Sox -- have working agreements with major league teams.

"If you've got a working agreement with the majors, its worth $90,000 to $100,000," said Mills, explaining that the big league teams traditionally pay player and manager salaries and some other expenses of their farm clubs.

Thomas has recently returned from Florida, where he said he had talked with the Los Angeles Dodgers, Texas Rangers and New York Yankees about the possibility of a working agreement between one of those teams starting with the 1979 season.

Thomas said that through the Dodger organization, the Carolina League will send Alexandria about 10 ballplayers. The club owners, who are mostly businessmen with lifelong interests in baseball, said they hope that two or three area players will make the team.

Thomas said he estimated it will cost about $175,000 to operate the team during its first season, and that the owners do not expect to make money from the venture but simply wanted to bring professional baseball back to the area. The owners plan to sell stock to the public with a minimum purchase of $100 required from each stockholder. The exact price per share has not yet been determined.

According to Mills, every team in the Carolina League made a profit last year, although the Salem an Winston-Salem teams made less than $5,-000 each. The Lynchburg Mets, who won the league championship, made the most money, with a profit of more than $17,000.

The 140-game Carolina League season is split into two parts, with the teams that win each half facing each other at season's end in a five-game championship series. Last year, Lynchburg beat Peninsula.

Opponents of an Alexandria team have argued that few fans would attend games played by a Class A team in an area accustomed to major-league baseball and other professional sports. Mills said attendance averaged about 800 per game last season, with the Lynchburg Mets attracting 55,123 fans for 68 playing dates, the most of any team in the league.

The playing field at Cora Kelly will have seats for 1,000 spectators with general-admission tickets expected to cost about $2.50. In other Itague cities, the general-admission tickets range from $1 to $2, but some teams charge as mych as $3 for box seats, according to Mills.