Delete the phrase “Let me know if I can help” from your vocabulary. Technically there’s nothing wrong with this phrase: it indicates a willingness to offer support. It also isn’t helpful in the least. When you tell someone to let you know if you can help, you put all of the work on them; they now have to identify where they need an extra hand, explain what you can do along with the how and why, and then communicate all of that back to you.
Look at the following examples:
• Not helpful: “Let me know if I can help with any of the applications.”
• Better: “There are three application deadlines coming up. Do you want me to look at these?”
• Best: “I noticed we have several application deadlines in the next month, and your plate is pretty full. I’m actually very interested in becoming more knowledgeable in this space, and due to my experience on X project, I’d love to prep these and bring them back to you for review this week so we can stay on target. Does that work for you?”
Let’s discuss why number three is ideal. The first one is still a version of “let me know,” so nix that. With the second you’re being slightly more specific but still asking for permission to do work in a way that’ll put the onus on your boss to figure out how to respond and what to do. In the third you’re indicating that you pay attention to detail and what work needs to be done, you’ve thought about why you want to work on this specific project, you offer an action-oriented solution that keeps your boss involved, and you reinforce the fact that this approach is efficient, productive, or good for the company.
Being specific and direct when you ask for an opportunity positions you for a “yes.”