Context
I am a mid level software engineer in California. I was contacted by a recruiter at an East Coast branch of a nationwide recruiting agency (not going to be more specific than that). Just earlier this week, the recruiting agency let me know that the four companies with which I've had phone interviews would like me to visit their offices for onsite interviews this coming week. Since this recruiting agency is acting as the point of contact for all four of these companies, I have not had any direct communications with them and have therefore not been offered to be flown out by the companies directly.
Just yesterday I had a phone call with the agency to discuss which days next week I would be able to go out there. During the call, the recruiter casually asked if I would be willing to foot the bill of travel costs. It was asked in a way that assumed I would be doing it. It caught me off guard and wanting to seem cool about it I said yes. Based on what I've heard and read, I'm under the impression that generally speaking software companies in the US pay for travel costs and so I assumed either one of the four companies with which I'm interviewing or the recruiting agency would cover the cost.
What I would like
I would like the recruiting agency to cover the cost of the flight (or at least partially reimburse me). I don't need accommodation or local transportation to be covered.
What to do
I feel like I should email the recruiting agency asking if they could either fully cover the cost or partially reimburse me.
Reasons I think I should ask
Unless I'm mistaken, it's just how interviews work. Companies/recruiting agencies pay for these things—it's part of the cost of recruiting.
My expected salary will be over 100k and the recruiting agency gets a certain percentage of my first year salary (somewhere around 15% I believe). I have made it clear that I am only interviewing with these companies and I will accept an offer if I get one. I think from their point of view they would be willing to take the risk of covering my traveling costs. It's just that they tried to minimize costs by seeing if I would pay for it.
If I don't get any offers I will have paid a lot of money just to have to start the interviewing process again.
Reasons I'm hesitant to ask
The biggest reason I'm hesitant to ask is that if they book it for me, they will probably find the cheapest option and I would lose flexibility in when and where I arrive/depart. I imagine they would fly me back Friday evening, however I want to fly back Sunday evening so I can stay with my girlfriend for the weekend. In my opinion this is enough of a reason to just bite the bullet and book the flight myself.
I believe they usually only recruit locally, so covering this sort of travel cost is not something they usually do. I don't even know if recruiting agencies usually cover the cost themselves (this is the first time I've dealt with a recruiting agency). The counterpoint to this is that they contacted me knowing I'm in California and that I would most likely have to fly out for onsites. Furthermore, they're a well-established nationwide agency—I think they should have the budget for it.
I already said I'd be willing to cover the cost and I'm afraid to tell them that I've changed my mind. This shouldn't be a reason, but I have to admit that I am easily affected by what others think of me. Of course, if I were to ask I could phrase it in a way that I was caught off guard when asked during the phone call and that I didn't have a chance to think it through and check how much the flight would cost. I would also not demand them to pay for it; the email would just be checking out what's possible in the way of making this less of a financial burden on me.