CAT and GMAT are accepted at IITs.
IIT Bombay and IIT Delhi are two good colleges for MBA.
IIT Delhi has CAT cutoff neaeby 96 percentile while IIT Bomvay has 97.2 percentile.
IIT delhi accepts CMAT/ SNAP/ MAT/ ATMA.
IIT Bombay has seen average package of 14lakhs per annum, so incase you're considering IITs for MBA, Bombay and Delhi are recommended.

For IIT Bombay MBA details check below for Placements records and intake.

http://www.som.iitb.ac.in/joomla/index.php?option=com_content&task=view&id=74

You need to sit for CAT in order to apply to the MBA colleges in IITs. You can apply to the MBA programmes offered by these colleges through their websites and then if you have a good enough percentile you might get an interview call from them.