Q: Forwarding imessages from phone to mac

Apple's Messages works via your Apple ID.  If you use yoru Mac's Applications -> Messages app and sign in using the same Apple ID, you will receive your Apple Messages on your Mac.

This will NOT forward your cellular carrier's SMS Text messages.

While an iPhone with iOS 8 or newer can forward SMS messages to a Mac running 10.10 Yosemite or newer via "Continuity", you need to have the iPhone on the same local area network (LAN) for that to work.

<Use Continuity to connect your iPhone, iPad, iPod touch, and Mac - Apple Support>

Hi,

It is a yes, No, Maybe sort of answer.

If the iPhone was only using the iPhone Number as the iMessages ID then the answer is no.

If the iPhone is lost you cannot add the Apple ID  the Mac might use on it's own to be able to link them.  (you can only add other IDs with physical access to both so that you can accept the pop ups that appear on the other device)

On the other hand if the Apple ID was added to the iPhone, the Mac version should be seeing al the iMessages sent to the iPhone Number and Apple ID that is in use (If you have both boxes ticked in Messages > Preferences > Accounts > Receive At list).

You have only listed an iPhone 5s and said you have a Mac without specifying what it is running - the presumption being as the Post is in Mavericks it is running a version OS X 10.8

This version is not capable of linking SMS Forwarding as well - but again this would also have to be done with the device (iPhone) in front of you. (As has been said the Mac needs be on Yosemite (OS X 10.10 ) and the iPhone on iOS 8 at least).

You may need to do this https://selfsolve.apple.com/deregister-imessage

As each device uses the Serial Number to "prove" it is your logging in getting a new phone and changing the number to the old one still may not let you have access until the first is deregistered.

This is unfortunately likely to lose the iMessages sent to to you in the mean time.

10:01 pm      Sunday; January 3, 2016