The Premier League club have been on the lookout for a new manager since Argentine Pochettino was appointed by Tottenham last month.
A host of names, most notably Ronald Koeman, have been linked with the post and Reed is confident a high-calibre candidate will soon take up the role.
"We are closing in now on our preferred candidate," Reed told Southampton's YouTube channel.
"When it was actually announced that Mauricio was leaving we were inundated with enquiries but by and large we were amazed by the level of applicant who wanted to show interest in the job.
"We had possibly six to 10 candidates on our ongoing research list. We were delighted that by the time we got down to the final four, all four of them had been on our list originally.
"Things are going well and I'm hoping – what takes time with these things is the paperwork – but I'm hoping that we can get that sorted very quickly and the fans will be excited about an announcement coming up sooner rather than later."
