The 22-year-old joined the Shakers in August 2014, following a brief loan spell at the club, and went on to make 81 appearances in all competitions.
Rose said: "[Moving to Mansfield] means playing games regularly, scoring a few goals, contributing to the team and hopefully a push towards promotion.
"It's a good club and I'm delighted to be here. Hopefully it will be a successful time here."
Rose scored five goals for the League One side last season, with his last league goals coming in the  4-3 victory over Blackpool in October.
The fee may be enhanced dependent on goals scored, promotion and a future sell-on clause provision.

The ruling by a US district judge is the latest twist in a long-running legal battle going back to 2013.
Costco, which is to appeal against the decision, argues that "Tiffany" is now a generic term for the rings.
But it has now been told it must call them "Tiffany-style" instead.
The dispute centres on the sale of solitaire-style rings, comprising a diamond mounted on a single band with six prongs.
Costco, which had sold 2,500 of them, put them on display with the label "Tiffany", although they were not in fact made by the jeweller.
An earlier court ruling in October 2016 ordered Costco to pay $5.5m in compensatory damages and $8.25m in punitive damages.
But in a further court decision on Monday, the compensation was increased to $11.1m, while the amount of punitive damages still stands.
Kate Swaine, a partner at law firm Gowling WLG, said: "This damages award may seem excessive given that only 2,500 products were sold, but if it can be argued that an infringement is blatant and where it relates to such a well-known brand, the claimant is entitled to seek punitive damages.
"Brand owners will welcome this decision as an indication of the risk that third parties run in trying to make associations with famous brands."
The cheapest engagement ring on Tiffany's US website currently costs $12,000 (Â£9,300) while customers can buy one from Costco's website for only $380 (Â£295).

On Monday the number of cars on the road in Paris will be cut in half - only motorists with odd-numbered plates will be allowed to drive.
Public transportation will be free as well car-sharing and bike-sharing programmes.
Similar measures were put in place last year when pollution soared in Paris.
The city saw a severe spike in smog on Wednesday last week and briefly had the world's dirtiest air, AP news agency reported quoting Plume Labs, a monitoring company.
The mayor's office announced at the weekend that alternating number plate measures would be implemented on Monday, with exceptions for electric or hybrid vehicles as well as emergency vehicles.
Public transport is free on Monday in Paris and surrounding towns to try to entice drivers to leave their cars at home.
About 750 police officers were being mobilised in the city and its suburbs to enforce the traffic restrictions which went into effect at 05:30 local time (04:30 GMT), Le Monde newspaper reported.
Other cities are also imposing various restrictions.
Mansfield have signed striker Danny Rose from Bury for an undisclosed fee.
[NEXT_CONCEPT]
US wholesaler Costco is facing a $19.4m (Â£15m) bill for damages after jewellery chain Tiffany sued it for infringing its trademark by selling "Tiffany" engagement rings.
[NEXT_CONCEPT]
The French authorities have introduced measures to combat a sharp increase in pollution affecting Paris and other cities in northern France.