Mr Law died last week after losing his battle with a brain tumour.
He snatched Blaenau Gwent, previously Wales' safest Labour seat, in
last year's general election after standing against his former party as
an independent. It followed Labour imposing an all-women shortlist on
the constituency.
A newly-formed political group, Blaenau Gwent People's Voice, has
already stated it will field candidates in both the Westminster and
Cardiff Bay seats Mr Law held. But it has emerged Mrs Law is expected
to throw her hat into the ring.
Mr Law's agent and friend, Dai Davies, had already said he would run
for the National Assembly seat.
Mrs Law, from Ebbw Vale, was not discussing her plans yesterday after
promising her children she would spend the day with them out of the
spotlight.
But Mr Davies said, although "no pressure" was being put on Mrs Law to
stand, she was "seriously considering it".
He said: "The fact is, Peter worked so hard to establish what he had in
Blaenau Gwent she doesn't want to see it lost.
"The situation is that the parties are destroying politics. Politics
belongs out there on the streets, it doesn't belong to the Tories or
the Lib Dems.
"In terms of socialism, that's what Peter believed and Trish supported
it. That's what Blaenau Gwent wants."
Mr and Mrs Law married in 1976 and had three daughters and two sons.
The woman who served as Mayoress of Blaenau Gwent in 1989 had
previously been the textbook politician's wife, appearing dutifully
alongside Mr Law on the campaign trail.
But she showed her feisty side during a TV interview on Thursday night
when she claimed her husband had been offered a peerage by Labour
chiefs in return for not standing against them, a charge the party has
denied. Police said they would look into the allegation.
Mr Davies added: "The only thing is she's never held any office at all,
but she's been alongside Peter all the way and will have picked up the
skills and knows what the score is.
"She can win, without a doubt. She's lived in and been part of the
community all her life. Her family are still there, her roots are
there."
No date has yet been set for the by-elections, although Labour scored a
PR own goal last week in attempting to move a writ in Parliament on the
day of Mr Law's death.
Labour are thought to believe an early election would benefit them.
Blaenau Gwent council leader John Hopkins will stand for the Assembly
seat, but their Westminster candidate has yet to be selected.
An Assembly by-election could be anything up to three months away.
Mr Law's funeral takes place on Thursday at Christ Church, Ebbw Vale.