Does twitter have something similar to facebook application invite pop-up?
In facebook you can write this:
FB.init({
appId : 'xxx',
status : true,
cookie : true,
oauth: true
});
FB.ui({
method: 'apprequests',
message: 'My Great Request',
data: '<?php echo $data; ?>'
}, requestCallback);
And it will pop-up dialog where you can select twitter friends, and send invitations to them, and then when invite is accepted with API you can find data to redirect them to site...
Is this somehow possible with twitter?
Does twitter have invite pop-up like facebook? or maybe sending multiple messages to selected users pop-up?
Twitter does not provide such an invite mechanism. You have two options.
Be carful however, because posting excessively to either will get your users and your application suspended.