Downes was expected to complete the loan switch last week, with James Ward-Prowse heading the opposite way in a £30million transfer.
The 24-year-old told Southampton's club website that he had hoped to be available to face Plymouth on Saturday, but food poisoning held up his arrival at the club.
'It's been a tough week, struggling with illness a little bit, but it feels good to finally be here, to get it over the line, and I'm just excited to get going,' Downes said.
'I was hoping to be involved on the weekend. Long story short, I had a bit of dodgy food off my own back – I'm not going to blame the missus.
Flynn Downes has completed a season-long loan move to Southampton from West Ham
Downes move to Southampton follows James Ward-Prowse's £30million transfer to West Ham
'I got food poisoning, felt like I was on my deathbed, lost a lot of weight, so it's not ideal, but it is what it is. I'm over it now, we live and we learn, I won't be eating raw chicken liver ever again.'
Downes will be reunited with Russell Martin, after he played under the Southampton manager at Swansea.
Downes played 39 matches for the Welsh club during the 2021-22 season with his performances convincing West Ham to sign him for £12million last summer.
The midfielder made 35 appearances during his debut season at West Ham, but made only seven starts in the Premier League.
He also made six starts during West Ham's run to the European Conference League title.
The midfielder admitted his desire for more first team football prompted him to make the move
Downes admitted that his season at Swansea was the 'best year' of his career, adding that he had spoken with West Ham manager David Moyes about his desire to play regularly this season.
'The way we [Swansea] played suited me down to a tee and I just found I slotted right in. I loved coming in every day – I think that's a big thing,' Downes said.
'It's a good watch, I'm sure all the fans are happy with it. It's a great style of play – what team doesn't want to dominate the ball like Man City? I think every team wants to be like that.
'I feel like I'm an all-round type of player. I suppose I like to do a bit of everything – I like to defend, pass the ball, win the ball back, dribble with it, so a bit of everything really.
Downes has been reunited with his former Swansea manager Russell Martin at Southampton
'I spoke many times with the West Ham gaffer and I came to the conclusion I wanted to play week in, week out. Obviously I know West Ham have signed a lot of midfielders, so I knew that would be tough, but I think a lot of things just appealed to me here.
'The ambition for the club is obviously to get promoted, to get back in the Premier League, and that enticed me a lot. You look at the squad we've got here, I think we've got a very, very good chance of that, so it's looking good.'
Downes has made 184 senior career appearances, scoring four times, across his time with Ipswich, Luton on loan, Swansea and West Ham.
Southampton have made a promising start to the Championship season, winning two and drawing one of their opening three matches as they target an immediate return to the Premier League.