Hey Trainline is just a reseller of normal train tickets of train companies. They do not sell reservations for Eurail. 

In your case, you are looking for the Frecciarossa High Speed Train Milano Central - Paris Gare de Lyon? This is not included in the Eurail Pass.

You can use instead the TGV High Speed Train between Milano Porta Garibaldi - Paris Gare de Lyon, they run at similar hours, because they are concurrent on this route. The reservation costs 35€, bookable via Eurail.com online or via SNCF Call center. Contact us by telephone | SNCF Press #85 for English, you will get the reservation as e-ticket via email.

P.S. look if it cost more the reservation for the TGV than a normal Ticket for the Frecciarossa (FR has low fares, because they are new on the line, so you can use 1 Eurail day for another trip)

Hello, 

I can reserve a seat of  Frecciarossa High Speed Train Milano Central - Paris Gare de Lyon from “https://partners.italiarail.com/results”  with 11 euro. 

The italiarail site says “ Passholder

This is a seat reservation and must be used in combination with a Eurail pass or InterRail pass that is valid for the entire journey. Travelers with rail passes are required to purchase passholder seat reservations for travel on Frecciarossa, Frecciargento, Frecciabianca, EuroCity, InterCity and InterCity Notte trains. Changes to date and time only are permitted. Changes to stations or route are not permitted. Seat selection and seat changes post order completion are not available. Type: Non-refundable.” 

Or is the reservation not valid?

I checked it now, and I don’t get further, when I click to buy it I get the error message: 

Train #9292: 190: No matching offered services - Please select a different departure time or Refresh Search to update the availability information.

So it does not work. 

Thank you for a prompt reply.  Yes, I am getting the error after putting my information. I guess I have to pay 50 euro for TGV then. Thanks. 

OR-if money is tight but time less so: there are far cheaper ALT ways-in the past people were so smart to look at that by themselves, but it seems that since all these planners on apps anyone has lost this capability. Take train via Swiss (beautiful views come extra for free)- to Basel-then local FR TER to Mulhouse or Strasbourg-then INland TGV to P-for maybe 10 or else 20€.

OR via Geneve and then hop over border Bellegarde/Annemasse for same TGV onward.

DO NOT USE THIS COMPANY. THEY STEAL YOUR MONEY. PASSES ARE USELESS FOR MOST TRAINS.