More than 1,000 members of the Public and Commercial Services Union (PCS) in eight locations are walking out in an escalation of the long-running spat.
The union said those who take action will be supported by a strike fund.
PCS Secretary General Mark Serwotka has written to the government calling for urgent talks to resolve the dispute.
He has accused ministers of treating their own workers differently from others in the public sector after negotiations were held with unions representing health workers and teachers.
The union is stepping up strikes, with a nationwide strike of more than 130,000 civil servants planned for April 28.
The Home office said the Passport Office has already processed more than 2.7 million applications this year, processing more than 99.7 percent of standard applications within 10 weeks, most of which are delivered to customers well within this timeframe.
There are currently no plans to change official guidelines stating it can take up to 10 weeks to get a passport.