Cook recently hinted that he could relinquish his position as captain following their Test series against India and confirmed that he will think about his future after a humiliating innings and 75-run defeat in Chennai sealed a 4-0 whitewash defeat.
Root is currently the vice-captain of the team, and Boycott talked up the importance of him gaining as much experience as possible if he is to lead England into the 2017-18 Ashes series in Australia.
"If we are to have a new captain, he needs all seven Tests to get his feet under the table. We don't want Alastair giving it up after three or four Tests," he is quoted as saying by BBC Sport.
"This hasn't just happened. It must have been on his mind for some time. After we've lost the series, he says Joe Root is ready for captaincy, and nobody forced that out of him. He's a top professional and a top batsman. But I don't think it's helpful to him or the team or English cricket if he's in two minds what to do.
"Whoever captains in England this summer has seven Tests [against South Africa and West Indies] before the biggest series of all in the Ashes. If it is going to be Root - which it will be - he needs seven Test matches to put his stamp on it. Players need to get with his style."
England have suffered a record-equalling eight Test defeats this year.