Is there a way to maintain a connection to a restful web service?

I'm working on an application which is a monolith. We have some features in our roadmap that I think would fit into a microservices architecture and am toying around with building them as such.

My problem: the application processes ~150 requests per second during peak times. These requests come in on raw TCP/IP connections which are kept alive at all times. We have very strict latency requirements (the majority of our requests are responded to within 25-50 milliseconds). Each request would need to consume 1 to many microservices. My concern is that consuming multiple restful web services (specifically creating/destroying the connection each time the service is consumed as well as TLS handshakes) is going to cause too much latency for processing these requests.

My question: Is it possible (and is there a best practice) to maintaining the state of a connection to a restful web service while multiple threads consume that web service? each request to consume the web service would be self contained but we would simply keep the physical connection alive.