Former Bournemouth boss Harry Redknapp, who managed the club from 1983 to 1992 and took them into the second tier of English football for the first time in their history in 1987, predicts Howe will be "back in the Premier League soon".
He told BBC Radio 5 Live: "I am very surprised. I felt sure he would look to get the club back up next season. The job he has done has been incredible.
"He took a club struggling in the third tier into the championship - which was a miracle - and then into the Premier, which I thought was impossible to do.
"Then for five years he had us playing incredible football, beating everybody at different times - it has been amazing.
"The stadium only holds 11,000 people, they had no right to get in the Premier League. For 100 years they were never out of division three or four. No disrespect but it would be like Rochdale, or clubs like that, coming into Premier League. It has been incredible.
"It will not be an easy job to follow him. He could not have done any more, and walks away with his head held high."