‘The eggs smell rotten.’ The verb in this sentence is-

Linking verbs are verbs that serve as a connection between a subject and further information about that subject.
They do not show any action; rather, they “link” the subject with the rest of the sentence.
The verb ‘to be’ is the most common linking verb but there are many others like become, seem, appear etc.
In the given sentence, ‘smell’ serves as a link between the subject “The eggs” and further information about that subject “rotten.”

Source: www.grammarly.com & examples.yourdictionary.com