I am travelling to New York in September. Thinking of apply for ESTA US visa. But unsure if I am eligible. My British Passport was issued in April 2005. Hence it doesn't have the chip all recently passports have. I am unsure if my passport is machine readable. Am I eligible for ESTA visa or can you please provide a phone no to clarify this information.
Since your passport was issued before the October 26, 2005 deadline, all you need is a machine readable passport (https://esta.cbp.dhs.gov/esta/WebHelp/ESTA_Screen-Level_Online_Help_1.htm#vwp4), it doesn't even require a digital photo. And according to Wikipedia, all British passports since ~1988 (though they may have issued old stock until 1993ish) have been machine readable so it would seem you are fine.
As @Gagravarr pointed out, the machine readable code is easy to identify. Should be a simple job to check.