If you ask for a higher pay based on an offer from another company, you're already screwing yourself over.
Your current job MAY accept your pay increase but if they do, there's already a break in trust. And you're setting yourself up for being replaced. You may as well take that new offer right now.
Better option is to negotiate based on skills, experience, responsibilities, and other factors.
That has been my strategy 4 times in the last 6 years. You can get a much bigger raise if you switch companies than if you stay and wait for a raise. If I had stayed at my first programming job, even with generous 10% raises every 12-18 months, my salary would only have increased by 50-70%. Instead, by switching jobs, I was able to get 20-35% raises each time, and now my base salary is 113% higher than it was 6 years ago.
Beware that this can come back to bite you. If I look at someone's LinkedIn profile and see a slew of jobs listed in months, not years, their resume is going in the (metaphoric) trash. I need to get at least 2 years out of people, preferably 3 (on the low end).
You just have to know when to slow down. The recruiter will recognize the pattern, and might fear you will leave at the first best opportunity. I always felt that 3-4 years with each employer is the most promising strategy (European job market).
Your current job MAY accept your pay increase but if they do, there's already a break in trust. And you're setting yourself up for being replaced. You may as well take that new offer right now.
Better option is to negotiate based on skills, experience, responsibilities, and other factors.