When I try to run the configure script with a openssl 1.1.0 directory specified as a dependency location, the config fails:

checking which TLS library to use... openssl
checking if a basic OpenSSL program works... no
configure: error: Unable to compile a basic program using OpenSSL

This contrasts with my OS-installed version of openssl 1.0.2, against which tcltls configures and compiles OK.

I use the following lines to compile with OS-installed openssl 1.0.2:

./configure --disable-stubs --with-tcl=<fulldir>/tcl8.6.8/unix --enable-deterministic
make tcltls.so

I then install and compile openssl 1.1.0 shared libraries locally without problems. I then try the following configure line for tcltls:

./configure --disable-stubs --with-tcl=<fulldir>/tcl8.6.8/unix --with-openssl-dir=<fulldir>/openssl-1.1.0 --enable-deterministic --includedir=<fulldir>/openssl-1.1.0/include

I get the error quoted at top.

My computer is running a standard updated install of Kubuntu 16.04

It seems to me that this has been fixed in the meantime.  Tcltls builds correctly on OpenSSL 1.1.1g.

This has been fixed with the changes to re-implement the TEA build system.
See commit [6c02d4d0291e784b] for the complete set of changes.