The important routes are the 19/19a connecting Shrewsbury and Telford town centres with Ironbridge and Atcham and the 576 route connecting Oswestry and Shrewsbury via the villages of Ruyton XI Towns, Baschurch and Bomere Heath.
Both currently operated by Arriva Midlands, the 576 will now be operated by Tanat Valley Coaches and 19/19a by Select Bus Services.
While Arriva bus passes will be accepted until the end of the year those with bus passes after that including academic year passes will have to request refunds from Arriva and buy new ones from the new operators.
Shropshire Council said both services will operate the same routes and there will be only minimal changes to the Ironbridge service times as previously done so by Arriva Midlands.
However there will be an extension of the Ironbridge route through Coalbrookdale to Telford Town Centre.
A council spokesperson said: "To avoid any confusion with similar numbered services operating in Telford, this service will be renumbered 96/96a. Customers using the service between Ironbridge and Telford are advised to check the timetable before travelling."
Any bus passes purchased through Arriva for use on these services will be accepted for the rest of this year.
"However, passengers will need to request a refund from Arriva for any annual or academic year passes as these will no longer be accepted after January 20. To request a refund from Arriva Midlands visit www.arrivabus.co.uk/midlands.
New passes can be bought directly from both operators.
Penkridge based Select Bus Services spokesperson, Ben Brown said: “We are very pleased to have been awarded the contract to provide the Monday to Saturday bus service between Shrewsbury and Ironbridge plus the extension of the services beyond Ironbridge to and from Telford Town Centre via Coalbrookdale.
"This is an ideal time for us to step up a gear to provide local bus services 96/96A which offer a lifeline for many communities along the routes and we will be pleased to see those regular customers continuing to travel on the new look 96/96A along with many new customers getting aboard this improved bus service."