Hi arvindasp,
Refer below sample code.
You need to make use of data-bs attribute and call the new bootstrap.Modal function for opening the popup.
HTML
<html xmlns="http://www.w3.org/1999/xhtml">
<head runat="server">
    <title></title>
    <!-- Bootstrap -->
    <script type="text/javascript" src='https://ajax.aspnetcdn.com/ajax/jQuery/jquery-1.8.3.min.js'></script>
    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.0-beta2/dist/css/bootstrap.min.css" rel="stylesheet" crossorigin="anonymous" />
    <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.0.0-beta2/dist/js/bootstrap.bundle.min.js" crossorigin="anonymous"></script>
    <!-- Bootstrap -->
</head>
<body>
    <form id="form1" runat="server">
        <div class="container">
            <center>
                <button type="button" class="btn btn-info btn-lg" data-bs-toggle="modal" data-bs-target="#MyPopup">
                    Open Modal</button>
            </center>
            <!-- Modal Popup -->
            <div id="MyPopup" class="modal fade" role="dialog">
                <div class="modal-dialog">
                    <!-- Modal content-->
                    <div class="modal-content">
                        <div class="modal-header">
                            <button type="button" class="close" data-bs-dismiss="modal">
                                ×</button>
                            <h4 class="modal-title">Select Fruit
                            </h4>
                        </div>
                        <div class="modal-body">
                            <asp:DropDownList ID="ddlFruits" runat="server" AutoPostBack="true" OnSelectedIndexChanged="OnSelectedIndexChanged">
                                <asp:ListItem Text="Select Fruit" Value="0" />
                                <asp:ListItem Text="Mango" Value="1" />
                                <asp:ListItem Text="Apple" Value="2" />
                                <asp:ListItem Text="Banana" Value="3" />
                            </asp:DropDownList>
                            <asp:Button ID="Button1" runat="server" Text="Button" OnClick="Button1_Click" />
                            <br />
                            <asp:Label ID="Label1" runat="server" Text="Label"></asp:Label>
                        </div>
                        <div class="modal-footer">
                            <button type="button" class="btn btn-danger" data-bs-dismiss="modal">
                                Close</button>
                        </div>
                    </div>
                </div>
            </div>
            <!-- Modal Popup -->
        </div>
    </form>
</body>
</html>
Code
C#
protected void OnSelectedIndexChanged(object sender, EventArgs e)
{
    Label1.Text = ddlFruits.SelectedItem.Text;
    ClientScript.RegisterStartupScript(this.GetType(), "Popup", "new bootstrap.Modal(document.getElementById('MyPopup')).show();", true);
}
protected void Button1_Click(object sender, EventArgs e)
{
    Label1.Text = ddlFruits.SelectedItem.Text;
    ClientScript.RegisterStartupScript(this.GetType(), "Popup", "new bootstrap.Modal(document.getElementById('MyPopup')).show();", true);
}
VB.NET
Protected Sub OnSelectedIndexChanged(ByVal sender As Object, ByVal e As EventArgs)
    Label1.Text = ddlFruits.SelectedItem.Text
    ClientScript.RegisterStartupScript(Me.[GetType](), "Popup", "new bootstrap.Modal(document.getElementById('MyPopup')).show();", True)
End Sub
Protected Sub Button1_Click(ByVal sender As Object, ByVal e As EventArgs)
    Label1.Text = ddlFruits.SelectedItem.Text
    ClientScript.RegisterStartupScript(Me.[GetType](), "Popup", "new bootstrap.Modal(document.getElementById('MyPopup')).show();", True)
End Sub
Screenshot
