Earlier this month, the club tendered the offer of a $17.8 million contract to the AL Cy Young Award runner-up.
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Under MLB rules, the offer is the "mean salary of the sport's 125 highest-paid players."
Cole informed the Astros on Thursday, which would have been the deadline to accept or decline the qualifying offer.
The Astros made the move in the event he signs elsewhere. Houston would receive a draft pick from Cole's new team as compensation.
The Astros can still continue to negotiate on a long-term deal with Cole.
Cole made $13.5 million in the past season in Houston. He could command a long-term contract north of $200 million, according to some analysts.
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