Mitchell yesterday confirmed Howe had turned down the opportunity to succeed Alan Pardew as manager of Cherries' League One rivals.
Howe is understood to have been interviewed by Saints chairman Nicola Cortese and director Les Reed after Mitchell had given permission for them to speak to him.
The 32-year-old had been linked with the post immediately after Pardew had been sacked 11 days ago, while Scunthorpe manager Nigel Adkins was also in the frame.
However, despite being short-listed and then offered the post yesterday morning, Howe politely declined before informing Mitchell of his decision following training at Canford School.
It was the second time in 10 months that Cherries had managed to ward off an approach for Howe following Peterborough United's interest in November.
Mitchell has struck up an excellent working relationship with Howe and assistant Jason Tindall since taking the reins at Dean Court 15 months ago.
He told the Daily Echo last night: If Eddie had decided to take the job, I think I would have had to consider my future as well.
I have accepted the fact that we are not going to be able to hold on to him forever and Southampton's interest was very flattering.
If they had been in the Premier League, where they were a few years ago, then I may have felt differently and might even have tried to persuade him to go because it would have been a good move for him.
However, although they could probably take him further in the long term, they are where they are and we are competing with them. I wouldn't have been doing my job that well if he had decided to go.
Mitchell added: I think it was the right thing to do. I am pleased for Eddie that he is being considered by bigger clubs and he is definitely one for the future. I think we will lose him eventually because he can go further than the club can.
Whether we were debt free and had a few million pounds kicking around, I don't think we would be able to compete in the Premier League because we just haven't got the fan base or the stadium to take the capacity to keep earning the sort of money that is needed to stay there.
Eddie would be foolish to turn down the sort of money he could earn in the Premier League and what he could achieve in his career by moving there. I don't think we could ever offer Eddie and Jason anywhere near what Premier League clubs could and I think that's where they will end up.
The Daily Echo has been unable to contact Howe.