The only way you could possibly get it done is with a custom transaxle, and even then I wouldn't give you very good odds.

The Supra is rear wheel drive. Its engine mounts lengthwise which attatches to the transmission, then the driveshaft, then the differential (which is in the rear suspension).

To put the motor into a front wheel drive car would require you to find some sort of transaxle (i.e. a transmission and differential in one unit) because you can't have front wheel drive without a transaxle. Front wheel drive cars (save for a select few) mount their motors sideways.

Nobody has made any sort of transaxle compact enough to stick on an inline six for a front wheel drive application save for Volvo in the S80 and XC90. Those two cars are much wider than an Integra, so you would need an even more compact transaxle than what Volvo produces. You could try to have someone make a custom bell housing on the Volvo transaxle but there's no guarantee you'd have enough room. Widebody kits do not expand the engine bay, they just make the wheel wells larger.

Another idea which I just thought up: the second-generation Acura Legend has a 3.2 liter V6 (not an inline six) that was mounted lengthwise and used a very weird transaxle to route power to the wheels. But now that I think about it a little more, that particular setup routed a driveshaft through the oilpan. You would most definitely not be able to adapt this particular transaxle from a Honda V6 to a Toyota inline six. There is also nowhere near enough room in the Integra engine bay to fit the Legend transmission.

This is not a simple issue of getting some mounts fabbed up and seeing if you can squeeze it in. You would need to have a completely new transaxle engineered from scratch for your application, and even then it probably still wouldn't fit. If by some miracle you manage to get a transaxle made, the car would be so hacked up by the time you fit the drivetrain as to be no longer streetable whatsoever.

Just to make sure you understand:

Your idea is basically impossible.