All commissions on Real Estate are negotiable. As a rule of thumb your Realtor should give you a break since he is getting 2 transactions out of the sale of your home. I have done it to my clients by lowering my commission on 1% . Talk to your Realtor and negotiate the reduction in commission. Good luck!
In my humble opinion, yes, you should. Your a captive audience for the agent who is going to get both sides of your business. In our market it's not uncommon for brokers to offer "move up" programs where they will discount the listing fee if you agree to use them to buy your new home. Since the broker will typically make a full buyer's agent commission on the buy side, they often discount their portion of the listing fee in this type of a situation. But remember, even if they typically charge say 6% for the listing commission, some portion, typically half, is going to get paid to the agent representing the buyer, so they can't just give up the full 6%.
Presuming the buyer's agent norm in your market is 3% and the typical listing commission is 6%, leaving 3% for the listing agent to earn on the sale of your house, and then another 3% for them to earn when you buy your new home, it's not uncommon to see a discount off the listing agent commission of half, or 1 to 1.5%. This is fair and reasonable for your agent. They will make 4.5% to 5% total.
It's not a bad idea to go hunting around your market by looking at broker websites and seeing if any of them have these "move up" type programs. Call several brokers and ask them what their policy is. After you do your research, present a recommendation to your agent and work out something that is amicable and fair.
Regarding negotiation, there's a saying that "The first person to mention price loses." The same might be said about negotiating commissions. Have in mind what might make sense to you. (Yes, I know that's what your question is asking. Think it through. You'll come up with a reasonable number.) Then pose that question to the Realtors you're interviewing. See what they offer. You might be surprised that they offer a commission reduction equal to or greater than what you were expecting.
Hope that helps.