Hi manvendra45,
Create DBContext with parameterized connection string.
So you can pass connection string before retrieving the data from DBContext.
Check the example.
Model
public class Customer
{
    public int CustomerId { get; set; }
    public string Name { get; set; }
    public string Country { get; set; }
}
DBContext
using Microsoft.EntityFrameworkCore;
public class CustomerDbContext : DbContext
{
    public CustomerDbContext(string databaseConnection) 
        : base()
    {           
        ConnectionString = databaseConnection;
    }
    public string ConnectionString { get; set; }
    protected override void OnConfiguring(DbContextOptionsBuilder optionsBuilder)
    {
        optionsBuilder.UseSqlServer(ConnectionString);
    }
    public DbSet<Customer> Customers { get; set; }
}
Namespaces
using System.Collections.Generic;
using Microsoft.Extensions.Configuration;
Controller
public class HomeController : Controller
{
    private IConfiguration Configuration;
    public HomeController(IConfiguration _configuration)
    {
        Configuration = _configuration;
    }
    public IActionResult Index()
    {
        return View();
    }
    [HttpPost]
    public IActionResult Index(string server)
    {
        string constr = this.Configuration.GetConnectionString(string.Format("MyConn{0}", server));
        return View(this.GetCustomer(constr));
    }
    public List<Customer> GetCustomer(string connectionString)
    {
        CustomerDbContext _contextFactory = new CustomerDbContext(connectionString);
        return _contextFactory.Customers.ToList();
    }
}
View
@addTagHelper *, Microsoft.AspNetCore.Mvc.TagHelpers
@model IEnumerable<Dynamic_Connection_Core_MVC.Models.Customer>
@{
    Layout = null;
}
<!DOCTYPE html>
<html>
<head>
    <meta name="viewport" content="width=device-width" />
    <title>Index</title>
</head>
<body>
    <form method="post" asp-controller="Home" asp-action="Index">
        <select name="server">
            <option value="">Select</option>
            <option value="Mumbai">Mumbai</option>
            <option value="Delhi">Delhi</option>
        </select>
        <input type="submit" value="Get Data" />
    </form>
    @if (Model != null)
    {
        <hr />
        <table id="tblFiles" cellpadding="0" cellspacing="0">
            <tr>
                <th>Id</th>
                <th>Name</th>
                <th>Country</th>
            </tr>
            @foreach (var customer in Model)
            {
                <tr>
                    <td>@customer.CustomerId</td>
                    <td>@customer.Name</td>
                    <td>@customer.Country</td>
                </tr>
            }
        </table>
    }
</body>
</html>
Screenshot
