External offer but interested in internal role

I have an external offer that is 20% more than my pay. I am torn on whether to take it. I want to try and negotiate and if they offer me more than I think I'll probably have to take it. The work isn't as interesting to me but the benefits and compensation and company are good.

But I recently saw an internal role on another team at my company that I really am interested in. I know multiple people on the project. I'm having an informal conversation with the hiring manager. If it goes well, I'm unsure how to approach the situation with the external offer timing. Do I tell my company that I have an external offer that I'm strongly considering but would love to stay if the process can be expedited?

Usually, in situations like this, the procedure for you, as a "worker", is:

• get a cancelable signed contract with the other company, with the salary and benefits clearly stated (among all the other contractual statements);
• go to your company and start negotiating. If they try to be tough, tell them about the other job, and the benefits there. Show that you have a backup plan. At this step, be ready that your company might want to get rid of you, and tell you good-bye. That s why you need a very good backup contract, not just a verbal discussion.
• make your decision about how to proceed - unless your company made the decision for you.

Bottom line:

• do not talk to your company about another company, unless you have a signed contract;
• you have to make the decisions for yourself, we cannot help - simply because we do not know all the relevant details.

I'm unsure how to approach the situation with the external offer timing

If the timing is tight, ask kindly the other company that you need more time before you make the final decision, as you negotiate with your current company. Show that you are loyal to the company where you work now, and tell the "other" company that you will be equally loyal to them if you will ever have to leave them - and offer them an opportunity to negotiate a mutually better contract. With other words, show that you want to be a win-win person.