I don't want to pass this variable from foo.pl

Why not? By accessing or manipulating the variable as a global variable it can be very difficult to determine where the value comes from and when or why it changes. By explicitly passing the value in as an argument to a sub or constructor it is much easier to manage and debug the code relating to it.

I don't want to pass the variable as an argument because this foo.pl is an existing object oriented perl script and uses a lot of modules. The "$variable" will be used in a sub-routine in one of the module which is called from few other modules. If I have to pass the variable as an argument I would need to make changes in a lot of perl modules and the top level script which I would like to avoid as far as possible. I tried $::variable in a small dummy perl program but it does not seem to be working for me. Regards.

Just a wild guess, but one conceivable reason that $::version would not work is that it's being used before it has been assigned its value. After all - as you have use Arithmetic; - the code is in an implicit BEGIN {...} block. In other words, in case there's any initialisation in the module, that code might be executed before $::version is ready.  For example, this would not work as one might expect at first:

#!/usr/bin/perl

use strict;
use warnings;

$::version = shift @ARGV;

BEGIN {  # emulate "use Arithmetic;"

  package Arithmetic;
  
  sub load_config {
      my $config = "/usr/local/myapp/$::version/properties";
      
      print "config: $config\n";
      # ...
  }
  
  load_config();
}
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In case you recognise any similarities to your scenario, you could try wrapping the assignment $::version = shift @ARGV; in a BEGIN block, which would make sure that things are being executed in the proper sequence.

Pass the variable into the constructor for the object that needs it then. Besides, a little refactoring up front to add such a facility rather than hacking it in is almost always worth while.

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Perl reduces RSI - it saves typing