From heatrow airport which part of middle east city is nearby to travel fast to bombay international airport?

I would say Tehran or Kuwait. Maybe even Bahrain and Doha. That is the furthest east you can go in the Middle East.

The further ones would include Riyadh and Dubai, but either way your looking at 7+ hrs of flying from the Middle East.

Why don't you just fly direct though?

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I think Muscat, Oman. But probably not that much difference in flight time between Muscat and Dubai. I would imagine Dubai would have more frequent service.

Save yourself time and money and fly direct!
British Airways have direct flights to Mumbai (BOM) Airport

Dubai and Doha are the best places for connectios when taking this route

Dubai, Doha, Oman
There are usually several flights a day to Bombay from any of these airports.

The following airlines fly LHR to BOM nonstop

Air India (Flt 144) 08hr 30min
Jet Airways (Flt 117 & Flt 119 ) 08hr 55min
Kingfisher (Flt 008) 09hr 00min

Turkish Airlines takes 11hr 05min, including 1hr 00min at Istanbul.
Oman Air takes 11hr 50min, including 1hr 50min at Muscat.
Gulf Air takes 14hr 00min, including 1hr 55min at Bahrain.

Results obtained from search-booking engines in the website referred below. (MakeMyTrip, Yatra.com & CheapOair)

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I think Dubai, Doha, Oman is too near