Lucas Silva is unlikely to join Real Madrid in January with the Spanish giants not planning to make any transfers.
Ravel Morrison has returned to parent club West Ham from Cardiff City on compassionate leave.
The 32-year-old has left Killie a year after joining and now has the chance to complete a League Cup double following his success at Hampden against Celtic in March.
The former Norwich player joins City in the midst of their Capital One Cup semi-final against Aston Villa with the npower League Two club 3-1 up following the first leg at Valley Parade.
Nelson wrote on his Twitter account: "Very pleased to have joined Bradford city fc. Hopefully have some great times ahead."
"Massive thank you to everyone at Killie."
"Nothing but great times and success since joining."
"Sad to leave but excited to join Bradford."
Kilmarnock, who are reported to be receiving £50,000 for the player, sold midfielder Liam Kelly to Bristol City last week.
Manager Kenny Shiels told www.skysports.com: "Michael will be a massive loss for Kilmarnock.
"If you look at 2012 then Kelly and Nelson were two of our best players. They were integral and a huge part of the success story of Kilmarnock in 2012.
"It is important that we look after the finances of the club as they were not players that I wanted to lose.
"We can't look a gift horse in the mouth and we have to look to the future. The money that we have got in for Liam and Michael is not a lot.
"Kilmarnock are a club that is massively in debt in Scottish terms. The debt we are in supersedes the size of the club and I'm really sad to lose both of those players."