I would weed it out unless you want grass in there. The sooner the better before it takes over and gets a real foothold. Pulling beats digging.

The new turf grass won't bother your OG's, but it will bother you. You won't be able to mow up close to the OG's, and that means extra trimming.

Eliminate the turf grass for at least a couple of feet around the crowns of the OG's. Mulch the soil after getting rid of the lawn grass.

You can put newspaper around the base of the OG's and cover the newspaper with mulch. I do that all the time. It kills whatever is growing under the newspaper and the newspaper ink contains zinc which prevents seeds from germinating. After a couple of years, the newspaper will disappear.