Expansion side FC Cincinnati selected UCLA midfielder Frankie Amaya No. 1 overall in the 2019 MLS SuperDraft on Friday.
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Beasley, 36, has played with the Dynamo since joining midway through the 2014 season.
The Houston Dynamo have signed Slovenian defender Aljaz Struna from Italian Serie B side Palermo, the team announced on Monday.
The MLS playoffs will be without Zlatan Ibrahimovic after the LA Galaxy lost to Houston on the final day of the regular season.
Nicolas Lodeiro assisted on all three goals to set a franchise career record as the Seattle Sounders defeated the host Houston Dynamo 3-2 on Sunday.
Carlos Vela scored his 12th and 13th goals of the season and added two assists as LAFC romped to a 4-2 victory over the Houston Dynamo.
Victor Rodriguez scored twice, helping the Seattle Sounders move within a point of fourth-place after a 4-1 win over the Houston Dynamo on Monday.
Houston's Romell Quioto made short work of a shot into the goalmouth that deflected off two defenders to score the deciding goal with three minutes
Mauro Manotas' brace ensured the Dynamo's first Open Cup title and the first trophy for the organization since winning MLS Cups in 2006 and 2007.
The USOC has at times been a refuge for an MLS club suffering through an otherwise forgettable season. So it proved for the Dynamo on Wednesday.
Mauro Manotas scored as the Houston Dynamo ended its 10-match winless streak with a 4-1 victory over the visiting Portland Timbers.
Santiago Mosquera scored a brace early in the second half to help FC Dallas roll to a 4-2 victory against the reeling Houston Dynamo.
Brian White's 55th-minute goal was the topper on an opportunistic performance by New York and lifted the Red Bulls to a 1-0 victory over Houston.
Ronaldo Pena scored the tying goal in the 88th minute to lift the struggling Houston Dynamo to a 1-1 draw with visiting FC Dallas on Thursday.
Albert Rusnak scored twice in second-half stoppage time to give Real Salt Lake a stunning 2-1 victory over host Houston at BBVA Compass Stadium.
Gyasi Zardes scored a minute into second-half stoppage time to give the Columbus Crew a 1-0 victory against the visiting Houston Dynamo on Saturday
The Philadelphia Union and the Houston Dynamo both advanced to the U.S. Open Cup final with semifinal wins Wednesday.
Diego Rubio scored in the 74th minute and goalkeeper Tim Melia turned away four shots as Sporting Kansas City did just enough to beat Houston.
Alejandro Bedoya, Cory Burke and Fafa Picault all scored as the Philadelphia Union rallied for a 3-1 victory over the 10-man Houston Dynamo.