Iam using following code to display a page select dropdown menu on an plugin option page.
If you select an page and save it, the next time you visit the option-site, you will also see which page was saved. (so no more “select page”)
You can use the WordPress selected() function, you can find more details here:
https://codex.wordpress.org/Function_Reference/selected
$options = get_option( 'my_settings' ); ?>
<select name="my_settings[selected_page]">
<option value="0"><?php _e('Select a Page', 'textdomain'); ?></option>
<?php $pages = get_pages(); ?>
<?php foreach( $pages as $page ) { ?>
<option value="<?php echo $page->ID; ?>" <?php selected( $options['selected_page'], $page->ID ); ?> ><?php echo $page->post_title; ?></option>
<?php }; ?>
</select>
You just need to get the current-saved-value of the field and than use the WP selected() function to compare the current-saved-value with the $page->ID.
Maybe this can help you.