Weeks, the second overall pick in the 2003 First-Year Player Draft, was forced to select new representation after Babby took over as the president of the NBA's Phoenix Suns.
The Brewers had a similar experience with Genske and associate Brian Peters in the leadup to the 2008 Winter Meetings in Las Vegas, where they finally got a sit-down and pushed hard for Sabathia to re-sign. Sabathia instead inked a richer deal with the Yankees.
Earlier this week, Melvin assigned the task of reaching Genske and determining Weeks' asking price to assistant general manager Gord Ash. Melvin expects to meet in person next week at the Winter Meetings in Lake Buena Vista, Fla.
Weeks, for his part, told MLB.com last month that he's open to the idea of an extension. Genske was not available on Thursday.
"I've had one meeting with [Genske], and he said that he's interested in a long-term deal," Melvin said. "But once players get close to free agency, they get a little less interested in trying to sign up, because they're so close to free agency and they've got some security under their belt."
Weeks earned $2.75 million in 2010 and had a breakthrough season. Finally healthy for a full year, he set career highs in games played (160), runs (112), hits (175) and extra-base hits (65), home runs (29) and RBIs (83).
He is arbitration-eligible one more time this winter before qualifying for free agency.
Melvin conceded that Weeks still has some defensive deficiencies, but credited the player for "becoming a much better second baseman."
"Is he a Gold Glover? No, he's not a Gold Glove-type, but he works hard at it," Melvin said in the radio interview. "He's athletic. He's a very good offensive second baseman, and there's only four or five. There's [the Yankees' Robinson] Cano, there's [the Braves' Dan] Uggla. There's [the Phillies'] Chase Utley. I think Rickie has probably passed Brandon Phillips as far as an offensive player. There's only about five or six good offensive-type second basemen.
"And the one thing about Rickie is that he can hit leadoff, he can hit three in the lineup, he can hit five in the lineup. He can hit wherever you [want] because of his skills."