How do I send telephone text messages from an iMac

one alternative is to use email to text from the iMac. Each particular phone provider may have a specific email address that will forward to the phone as a text message, for example for AT&T phones (even non-Apple phones):

https://www.att.com/esupport/article.html#!/wireless/KM1061254

"Send a message to an AT&T wireless device by email

• Text message: Compose a new email and use the recipient's 10-digit wireless phone number, followed by @txt.att.net. For example, 5551234567@txt.att.net.
• Picture or video message: Compose a new email and use the recipient's 10-digit wireless phone number, followed by @mms.att.net. For example, 5551234567@mms.att.net.
• Don't include dashes or spaces in between the numbers."

Send text messages to anyone from your Mac:

If you have an iPhone with an active cellular plan, you can set up Messages to send and receive SMS and MMS messages from your Mac. Such messages don't require the recipient to be using iMessage or an Apple device.

1. On your iPhone, go to Settings > Messages > Send & Receive. Add a check to your phone number and email address.
2. On your iPhone, go to Settings > Messages > Text Message Forwarding, then choose to allow your Mac to send and receive text messages from this iPhone.
3. Check your Mac for a verification code, then enter that code on your iPhone.

To get all of your iMessage and SMS/MMS messages on all of your Apple devices, you can set up the SMS and MMS feature of Continuity.

You should be able to use Messages in your Mac for standard SMS text to non-Apple device

users. These may accrue a fee or simple charge. Some plans have latitude to not impact the

monthly costs, while others may accrue noticeable charges, depending on other use factors.

Roaming, out of country, etc may be at a premium to certain levels of cellular plan service.

• Use Messages with your Mac - Apple Support

Learn the difference between iMessage and SMS/MMS.

[To use SMS/MMS on an iPhone, you need a text-messaging plan. Contact your wireless carrier for more info.

You can also set up your other Apple devices to send and receive messages from any Apple device.]

For these features to work, the iOS and macOS have to be within reasonable vintage

and the iPhone (or iPad with Cellular+Wi-Fi) within localized proximity of your Mac.

Good luck & happy trails! 🙂

There are settings that have to be properly selected, for this to work.

And key, is the iPhone; you have to choose careful; this is what does

the cellular access, even when turned off (in near proximity) once set.

Then the Mac can use Messages application to send SMS to third-

party smart phones. Sometimes even when correctly set, a service

provider may have a problem that interferes with transmission.

Depending on your iPhone's iOS version, your Mac's OS X or macOS

version, and your cellular data plan... the iPhone will use wi-fi locally

to stay in touch with the Mac (or with a wi-fi iPad) etc to share content.

Each of the previously posted linked articles have bits of information

So if there was a problem with implementation, you may need to ask

your cellular provider ~ or arrange a call back from Apple support.

At distance without seeing your different products or their settings, I can

say that even with them set correctly, there are times when I cannot tell

if or when my Messages (Mac Mini i7 Late2012 quad server, runs 10.11.6

w/ iPhone 5C iOS10.3.3) has completed a Message to non-Apple users.

Could be the settings in the users devices interfere with the action or if

their own cellular provider was down; if spotty performance is indicated.

An interactive step-by-step and then back-tracking may be necessary to

see if your stuff is setup correctly. And as a volunteer non-employee and

a fellow product user such as yourself, I have no extensive resources.

In any event...

Good luck! 🙂

"...for example for AT&T phones..." -- glen doggett

Generally your setup of SMS and MMS if done correctly, should route your messages.

Set the Contacts in iPhone with each persons info (email, phone, name, etc) as that would

be shared via iCloud with your Mac; & check Contacts in your Mac to see if they got there.

If your iPhone isn't set correctly + your Mac also lacks the setup steps done, it won't work.

And at times, even if these are correct, sometimes results are spotty due to carrier or wi-fi.

Good luck & happy trails! 🙂