If you don't have a strong BATNA (Best Alternative to a Negotiated Agreement), you're not going to get very far with just asking.
"I received your offer for $X. I really enjoyed the interviews, meeting the team, and am intrigued by the problems I'd get to solve with them. Unfortunately, I have a competing offer for $1.25X from another company and I need to make sure my family is provided for. If you would match it, I'm excited to join, but if your offer is firm at $X, I'm afraid I won't be taking it."
If they really just need someone to hold down a chair, they'll probably not budge. If they really want specifically you, they're more likely to bend if they believe that's their only move.
"I received your offer for $X. I really enjoyed the interviews, meeting the team, and am intrigued by the problems I'd get to solve with them. Unfortunately, I have a competing offer for $1.25X from another company and I need to make sure my family is provided for. If you would match it, I'm excited to join, but if your offer is firm at $X, I'm afraid I won't be taking it."
If they really just need someone to hold down a chair, they'll probably not budge. If they really want specifically you, they're more likely to bend if they believe that's their only move.