First of all you can send session variable. you will have to embed them with your url, but you can't automatically expect that your session to get automatically.
the possible solution, as example:
<table >
<tr>
<td >
<a href='http://www.mywebsite.com/profile-page.php?reg_id=$r1&username=<?=$_SESSION['user_name']?>' target='_blank'>Open Profile</a>
</td>
</tr>
</table>