I can't give you an answer to this but you may wish to refine the question depending on the purpose you have for the information. By 'the exact number of bird species' do you mean regular breeding species, or to include occasional breeders, winter visitors, passage migrants and even vagrants that have occurred just once or twice? The biological impact of these (or other) segments of the Indian bird list will all be different. And then there are differences in taxonomic approach, just to make things more complicated. 'Exact' is rarely an exact science, unless this is for competitive bird-listing!
By exact i mean the numbers of bird species which we can use as a references for paper to compare what we observe during the field visits.
By bird species' we mean regular breeding species, including occasional breeders, winter visitors, passage migrants and even vagrants that have occurred just once or twice but are mentioned in literature.
e-bird India carries an article on names (taxonomy) and what you may be interested in - a list of species. Do see if that works for you. http://www.birdcount.in/ebird-names-for-indian-species/
There does not appear to be an acceptable list of Indian birds currently. The eBird checklist specifically excludes species with less than 10 records. The Avibase & BirdLife lists contain quite a few errors - both excluding species with valid records and including species that have not been recorded. And the taxonomy used by these three lists does not correspond so comparison is difficult.