To promote, praise, or advertise someone or something in an extravagant or overly enthusiastic manner. A noun or pronoun can be used between "tout" and "around." There are scores of nutrition companies touting around their supplements as miracle cures for any number of illnesses and ailments, but they're all just a bunch of snake oil. The boss has been touting the new vice president of marketing around like she's going to be the savior of the company.
tout someone or something around
to promote and boost someone or something publicly. He is touting his favorite candidate around, hoping to get a few votes for her. Roger is touting his book around, trying to boost sales.