The former Cuba U23 international, who has since defected to the US, will train with the Dynamo until Thursday and then the club will see what options they have to sign him.
"We played Austin in the preseason and then went up and saw them against Montreal in the playoffs," said Dynamo head coach Dominic Kinnear. "We circled some names that we thought were pretty good players and, speaking with the coach, there was an opportunity to bring him in.
"We're fairly certain he's under contract with Austin but the way [Aztex head coach] Adrian Heath has gone about business, he has been very fair especially with us."
Alvarez admitted he was enjoying his time at the club and said he was learning plenty in an elite environment.
"I felt a little tired because I haven't been training regularly," he said. "I felt good and confident, though. When my physical fitness improves, I will be better."