The Los Angeles Lakers have narrowed their coaching search to former Cleveland Cavaliers head coach Mike Brown and former Houston Rockets coach Rick Adelman, according to sources, with current Lakers assistant Brian Shaw, the preferred choice of most of the team’s current players, still in the mix. But the Lakers’ insistence on holding firm to a lower salary for their next head coach has kept the search open.

The Lakers want to significantly reduce the salary they pay their next head coach after paying Phil Jackson $10 million in his final season and $12.5 million the previous season. The Los Angeles Times reported Monday that the Lakers were not willing to go above $5 million in salary next season. While Brown is not expecting to make $7 million a year as Doc Rivers will be getting from Boston starting next season (Rivers signed a five-year, $35 million deal last week), the next tier of coaches, led by New York’s Mike D’Antoni (four years, $24 million), don’t have Jackson or Rivers’ pedigree, while Brown has a 2007 Finals appearance and two Eastern Conference finals appearances to his credit.

But the Lakers are adamant about maintaining their salary ceiling, and seem to be looking for the candidate that will accept those terms as opposed to picking a candidate and negotiating a deal with him.

Yahoo! Sports reported Tuesday evening that the Lakers were in serious negotiations with Brown, who has spent this year working in television with ESPN after being fired as coach of the Cavaliers after Cleveland’s second-round loss to Boston in the 2010 playoffs.

Lakers owner Jerry Buss had said earlier in the day in a radio interview on XM Sirius Radio that the team was “very close” to hiring a new coach, and seemed to dismiss the notion that the team would hire Shaw because several players, including Kobe Bryant and Derek Fisher, had endorsed his candidacy.

“Obviously, we have to select somebody who has a reputation that players would be happy with,” Buss said in the interview, on Playboy Radio, with local broadcaster Michael Eaves and Lakers executive Bonnie-Jill Laflin. “But to ask a direct player to select a particular coach, that’s general manager territory.”

Brown, 41, was 272-138 in five seasons in Cleveland, reaching the second round of the playoffs each season. But after the Cavaliers’ six-game loss to Boston in which the Cavs played poorly in the last two games, Cavaliers owner Dan Gilbert decided to fire Brown. That decision helped lead to former team president Danny Ferry’s departure from the club as well, and James, famously, followed both out the door in July.

Adelman, 64, moved into eighth place on the NBA’s all-time coaching victories list this season, finishing the year with 945 wins, one better than Bill Fitch and seven ahead of Red Auerbach. In 20 seasons as a head coach, Adelman is 945-616 (.605), with only four losing seasons. His teams have made the postseason 16 times, and he’s reached the Finals twice, both times with Portland (1990 and 1992). This season may have been one of his best coaching jobs; Houston had hoped to have Yao Ming back this season after he’d missed all of the 2009-10 season with a stress fracture in his left ankle, but Yao only played five games before suffering another stress fracture in the left ankle that shelved him the rest of the year.

But Adelman remade the Rockets on the fly, turning them from a team that was planning to pound the ball inside to one that got its scoring from the perimeter with guards Kevin Martin and Kyle Lowry. Houston didn’t make the playoffs, but was one of the hottest teams in the league down the stretch, going 17-8 after the All-Star break. Nonetheless, the Rockets made next to no attempt to re-sign him and the club announced after the season it and Adelman had mutually agreed to part ways.

Shaw has been a Lakers assistant for five years, gaining the respect of players as someone who can speak candidly to players and challenge them. He played with the team from 1999 to 2003, helping the team win three straight NBA titles. He has said that he would maintain the triangle offense that Jackson used both in Los Angeles and Chicago, though he would tweak it some. Buss said that he expected the team to play some triangle next season but also expected some changes in the offense.

The Lakers have moved away from veteran coaches Jeff Van Gundy and Mike Dunleavy, the ex-Clippers coach and general manager who led the Lakers to the 1991 Finals.