No, but what you might try is writing a subclass of DBI:

package myDBI;

use DBI;

our @ISA = qw(DBI);
sub connect {
 my $proto = shift;
 my $class = ref($proto) || $proto;
 my $dbh = $class->SUPER::connect(@_);
 bless $dbh, "${class}::db";
}

package myDBI::db;

our @ISA = qw(DBI::db);
sub prepare {
 my $dbh = shift;
 my $sth = $dbh->SUPER::prepare(@_);
 $sth->{private_MyDBI_dbh} = $dbh;
 $sth;
}

package main;

my $dbh = myDBI->connect('dbi:Drivername:dbname',
    'user','passwd',{RaiseError=>1});
my $sth = $dbh->prepare("select stuff from table");
my $dbh_copy = $sth->{'private_MyDBI_dbh'};
[download]

Then you would also have to write a prepare() method to save the db handles in the statement handles using a 'private_*' attribute name (see the DBI docs). You could make it transparent except for the initial connect. Don't ask me if this'll work though, I've never tried it :)

Update: Fixed code. DBI->connect actually returns a DBI::db object. Oh what the heck, I wrote the prepare method also. All completely untested.