I was once an intern at a company, and had good relationship with my supervisor. I recently asked him for a reference letter and he said he would write one for me, with no hesitation. However, ten days have passed. How do I remind him about this issue without offending him?
I've been in this situation a couple of times back in the day. My solution is pretty direct. I save them the effort and check all the dates and everything else, then write the reference myself. Then I take it to them to get signed or discussed and use it as a good reason to visit and have a catch up, perhaps even bring a small cake or something for the team (makes a nice impression to everyone).
This makes the whole thing easy and pleasant for them. A bit more work on my side, but I'm the one who benefits so fair enough.
All they have to do is sign it or make small changes and then I'm off with everyone wishing me well.
How to respectively ask him about this issue without making him feel even the slightest sense of unhappy?
A phrasing I can think of goes something like this. Feel free to adapt it to your own style and relationship with your old boss and locale details:
Greetings, [old-boss], hope you are doing well and everything is going smoothly at the company.
I'm writing this email as a follow-up to the Reference Letter that we discussed a week ago, and that you kindly offered to procure for me. I was wondering if you had the time to write it? I know you are very busy, so I don't want to interrupt if you are in the middle of something critical, but my deadline for submission is closing and I would like to have everything ready before that happens.
Again, thank you very much for the Letter and your time. Cheers,
Perhaps this phrasing is a bit "too respectful", if such thing exists hehe, so again feel free to adapt it to your own tone, and style, as well as any cultural and locale aspects that are the norm where you are located.
Alternatively, if your old boss has an assistant or similar, consider reaching out to them to check if the letter is available and follow up. If they have no idea or are unaware of, then writing to old boss would be the next logical step (although you could straight email your boss, depending like I said on your relationship with them, cultural and locale aspects, etc.).
Edit: As Joe mentioned in comments, another option is to pick up the phone and call them, and politely ask if they were able to finish the letter. This option, if available or possible, is surely way faster than an email approach. This can also apply to the assistant or similar (picking up the phone and calling them, inquiring if old boss finished the letter).
However, IMHO, calling someone can be more "abrupt" or "disruptive" than writing an email, although not something that any reasonable person/boss would be mad or unhappy about (even less likely if they offered you something and you are reaching to follow up).