Here are the options for virtualenv

$ virtualenv
You must provide a DEST_DIR
Usage: virtualenv [OPTIONS] DEST_DIR

Options:
  --version             show program's version number and exit.
  -h, --help            show this help message and exit.
  -v, --verbose         Increase verbosity.
  -q, --quiet           Decrease verbosity.
  -p PYTHON_EXE, --python=PYTHON_EXE
                        The Python interpreter to use, e.g.,
                        --python=python2.5 will use the python2.5 interpreter
                        to create the new environment.  The default is the
                        interpreter that virtualenv was installed with
                        (/usr/bin/python)
  --clear               Clear out the non-root install and start from scratch
  --no-site-packages    Don't give access to the global site-packages dir to
                        the virtual environment
  --unzip-setuptools    Unzip Setuptools or Distribute when installing it
  --relocatable         Make an EXISTING virtualenv environment relocatable.
                        This fixes up scripts and makes all .pth files
                        relative
  --distribute          Use Distribute instead of Setuptools. Set environ
                        variable VIRTUALENV_USE_DISTRIBUTE to make it the
                        default
  --prompt==PROMPT      Provides an alternative prompt prefix for this
                        environment

1) What you want to do is install python to a directory that you are able to write too.

You can follow the instructions here.

For Python 2.7.1
Python source

mkdir ~/src
mkdir ~/.localpython
cd ~/src
wget http://www.python.org/ftp/python/2.7.1/Python-2.7.1.tgz
tar -zxvf Python-2.7.1.tgz
cd Python-2.7.1

make clean
./configure --prefix=/home/${USER}/.localpython
make
make install

2) Install virtualenv
virtualenv source

cd ~/src
wget http://pypi.python.org/packages/source/v/virtualenv/virtualenv-1.5.2.tar.gz#md5=fbcefbd8520bb64bc24a560c6019a73c
tar -zxvf virtualenv-1.5.2.tar.gz
cd virtualenv-1.5.2/
~/.localpython/bin/python setup.py install

3) Create a virtualenv using your local python
virtualenv docs

mkdir /home/${USER}/virtualenvs
cd /home/${USER}/virtualenvs
~/.localpython/bin/virtualenv py2.7 --python=/home/${USER}/.localpython/bin/python2.7

4) Activate the environment

cd ~/virtualenvs/py2.7/bin
source ./activate

5) Check

(py2.7)$ python
Python 2.7.1 (r271:86832, Mar 31 2011, 15:31:37) 
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> exit()

(py2.7)$ deactivate
$ python
Python 2.6.6 (r266:84292, Sep 15 2010, 15:52:39) 
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> 

Pre-requisites:

1. sudo easy_install virtualenv
2. sudo pip install virtualenvwrapper

Installing virtualenv with Python2.6:

1. You could manually download, build and install another version of Python to /usr/local or another location.

2. If it's another location other than /usr/local, add it to your PATH.

3. Reload your shell to pick up the updated PATH.

4. From this point on, you should be able to call the following 2 python binaries from your shell python2.5 and python2.6

5. Create a new instance of virtualenv with python2.6:

   mkvirtualenv --python=python2.6 yournewenv

Now a days, the easiest way I found to have a more updated version of Python is to install it via conda into a conda environment.

Install conda(you may need a virtualenv for this)

pip install conda

Installing a new Python version inside a conda environent

I'm adding this answer here because no manual download is needed. conda will do that for you.

Now create an environment for the Python version you want. In this example I will use 3.5.2, because it it the latest version at this time of writing (Aug 2016).

conda create -n py35 python=3.5.2

Will create a environment for conda to install packages

To activate this environment(I'm assuming linux otherwise check the conda docs):

source activate py35

Now install what you need either via pip or conda in the environemnt(conda has better binary package support).

conda install <package_name>

The usual approach is to download the source and build and install locally (but not directly in virtualenv), and then create a new virtualenv using that local Python install. On some systems, it may be possible to download and install a prebuilt python, rather than building from source.