No it is not possible as by definition the request occurs within a session and that session is bound to a TCP stream which is itself bound to an IP address. What you are asking for is to change the IP address of the binding which would break the connection.
But that's a hard "no" only because of how you framed your question, as what you are asking for shows a poor understanding of how web browsing works at a network level.
There are many ways to achieve similar effects. Direct server return allows the server to return results without passing the results through the load balancer. This has tremendous efficiency for web servers as only the inbound traffic needs to pass the load balancer. Outbound traffic can be sent directly. Since web traffic generally has small requests generating large results this model allows a relatively smaller load balancer handle relatively larger web sites.
Most load balancers work at high layers of the network stack processing high level HTTP streams. This is not a requirement however so you could pick a lower level load balancer that works at layer 2 or layer 3 to balance traffic across IPs which multiple servers can handle directly. In this model every server is effectively it's own load balancer so there is no middle-man dedicated load balancer processing traffic at all.
There is no particular reason to hold only a single session. You may logon to one load balanced server and pass the client a page that loads content or redirects to another server (or group of load balanced servers). In particular if you use IFrames properly the content of different frames can appear as if it came from a single server. Similarly you might use multiple systems that are optimized for multiple roles. Very commonly this is setup as a content distribution network in which a smaller number of servers handle dynamic content while typically a larger number of distributed servers handle static content.
There are a wide range of solutions to this problem. In general you would want to research options for scaling out web farms. But your question is flawed as the load balancer server will never allow the client and the destination server to "meet"/to "know each other". From the perspective of the client there is only one server and the client will never know anything about the infrastructure behind the load balancer.